Does modern turkish cooking have its roots in the Byzantine Empire, or the Ottoman Empire?

It's worth noting that in Turkey today, Osmanlı Mütfağı ("Ottoman Cuisine") is differentiated from "Türk mütfağı". To get a sense of Ottoman cuisine, look at my favorite Ottoman restaurant in Istanbul, Haci (a.k.a Hajji) Abdullah. It's so good... but it's pretty different from contemporary Turkish cuisine, which is much simpler than this.

It's also worth noting that while many dishes are popular through the country, Turkey is considered to have seven regions, each more or less with at least a few regional specialties. There are some pretty clear differences between the regions. For example, the four coastal regions rely on seafood a lot more (obviously), but also tend to have a lot more vegetables. Black Sea cuisine is famous for its ubiquitous use of hamsi (a small, anchovy like fish), but it can also feature a lot more cheese than Central Anatolia; for example, along the Black Sea there's a fondue like cheese, butter, and corn meal mixture (it's known by many names, including kuymak, muhlama, and yağlaş). For traditional Black Sea food, check out the restaurant Klemuri (though it also offers non-Black sea foods, like Hummus). Here, you'll see many dishes that are similar to (Pontic) Greek, Georgian, and Armenian food, as they all lived in close proximity, and Turkish Wikipedia tells us that Dagestan, Georgia, and other regions of the Caucasus. I know little about the history of those cuisines, however, so it's hard in many cases to say who influenced whom. With kuymak, we have at least etymological evidence. The first usage of kuymak to refer to a dish, however, Sevan Nişanyan (the best Turkish etymologist) tells is from ~1070, where it refers to "a bread that is made by pouring very hot oil/fat into a flour and milk mixture", meaning that I'd be comfortable guessing that this dish has Turkic rather than Greek, Armenian, or Georgian roots. Kuymak is, incidentally, an Old Turkish word for "to pour". To put 1070 in chronological context, the Ottoman dynasty wasn't founded until 1299, and didn't conquer Istanbul until 1453. In fact, this is the same period that the Battle of Manzikert (1071), which was the Byzantine military loss that allowed the gradual Turkification of Anatolia. The term "hamsi", on the other hand, is clearly not Turkic, and is taken probably directly from the Greek χάμψι (χámpsi), though Nişanyan notes that the Greeks could have themselves borrowed the word from an even older Black Sea language. But basically, you have to go not only region by region, dish by dish.

The Aegean region, on the other hand, has both heavy Greek influence (many dishes are recognizable on both sides of the Aegean Sea, just with separate Greek and Turkish names) and a heavy Caucasian influence (as in Georgia, Armenia, Ossetia, etc), because of the large numbers of Muhacir ("refugees") coming form the Russian ethnic cleansing of the Caucasus in the 19th century who were settled in the region (to counter-balance the very large ethnic Greek population, and after 1923, replace it completely). An Istanbul restaurant that represents this Aegean style of cooking is Ficcin. Unfortunately, the website isn't English, but many of the seafood and vegetable dishes have Greek counterparts. However, their two signature dishes are both Circassian--if you look at the first picture on restaurant's webpage, the slice of meat-filled crusty pie in the center is fıccın (clearly a non-Turkish word) and then directly in front of it are Çerkez mantı--literally Circassian mantı. Well, what are mantı? Mantı are a type of dumpling found from the Aegean to Central Asia and even beyond to Thailand, Japan, China, and the Philippines--the name ultimately derives from either Mongolian or Chinese, though it seems more likely that it's ultimately from Mongolian (or another related Central Asian language). In Turkey, the most common version is Kayseri's, which are much smaller than Circassian mantı. But this kind of mixing and widespread influence of food is very common.

We see people adopting foods as they move into a new area, people bringing their traditional foods with them, and trade routes spreading food as well. As far as I can tell, every region has been affected by all three. Just as some Turkish food was clearly Greek or Arab before the Turks arrived in Anatolia, some Greek and Arab foods are almost certainly originally Turkish. Gyro and shawarma are both calques of the Turkish döner, with all three being related to the verb "to turn" (though I'm told shawarma actually originates from another Turkish word meaning "to turn", çevirme, rather than an Arabic root). And this not even a particularly recent Turkish invention--döner originates in Bursa in the Marmara region in the 19th century. The 19th century innovation was essentially having the technology to turn an Eastern Anatolian, horizontal-spit çağ kebab vertically upright. Çağ kebaps apparently are mentioned in travelers' accounts at least as far back as the 18th century. Cacık is a yogurt dish that's of unclear origin, but the word is certainly Kurdish or Persian. Nişanyan says the Greek name, tzatziki, seems to be taken from the Turkish, while the Armenian, cacıχ, may have been taken directly from the Kurdish.

I'll just mention a few more major dishes that help illustrate the diversity of origins of modern Turkish food. Everyone seems to agree that the friable cheese called helloumi in English is called "hellim" in Turkish, "χαλλούμι" (halloumi) in Greek, "حلوم‎" (ḥallūm) in Arabic originates in Cyprus, and probably while the island was under Byzantine control (definitely before the Turks arrived). However, the word itself is unambiguously Arabic, the word meaning soft, though it's unclear how the word got to Cyprus (whether from Egypt, as OED claims, or Syria, as Nişanyan claims), and whether the term was merely applied to this Greek Cypriot cheese, or whether the cheese itself has some Arab or Semitic roots.

Yogurt is unambiguously Turkic in etymology and invention, going all the way back to Central Asia. Rakı, the national tipple, is from the Arabic word meaning "sweat", ˁaraḳ, and since it's a distilled liquor, could easily be a considered an Arab drink. The yogurt drink ayran, similar to savory kefirs or lassis, is likely of similar Central Asian Turkic origin (our first reference to it is similarly ~1070).

Pide, a genre of popular pizza-bread like dishes (the two most popular being the large, open-faced "Black Sea" Karadeniz pidesi, and the small, soft, closed-faced "field" kır pidesi) are etymologically and historically related to pita, the flat bread we know and love, and indeed to pizza itself. It's likely that the word entered Turkish from Greek, but etymologically it seems ultimately derived from Aramaic/Syriac, so that's another possible source as well as Arabic.

Kebab (kebap), probably the core of modern Turkish cuisine, ultimately comes from Arabic word meaning "fried" or "friend meat". Interestingly, mangal or grill, the way that kebabs are traditionally cooked, is also an Arabic word (which doesn't mean the Central Asians had no grills--many, many Arabic and Persian words replaced even basic Turkish words. The contemporary word for fire, ateş, is Persian, having replaced od/ot centuries ago). The particular kind of kebab most popular in Turkey, the Adana Kebab, was traditionally made in a broad swath that included the now Turkish cities of Mersin, Adana, Urfa, and Antakya, the Syrian Aleppo, and the Iraqi city of Kirkuk, which lies on the border of Kurdish and Arabic speakers, and has also long had a very substantial Turkic-speaking population. In Turkish novels from the 1920's I've read (like Orhan Kemal), this seems to have been THE eating out dish, as a surprising amount of the action takes place in kebab shops. Köfte, Turkish-style meatballs commonly also called koftah, comes from the Persian word for "beaten", "broken". The first usage of this loan word is quite early, ~1360, making me suspect that this was possibly borrowed even before the Turks reached Anatolia. Most Turkish words for Islamic rituals, for example, come from Persian rather than Arabic, because the Turks learned "civilization" (including Islam) from the Persians.

Turkish salads (salata) are very old, and etymologically the word comes to Turkish from one of the Italian dialects (there were many Levantine Italian traders in both the late Byzantine and the Ottoman Empires), but ultimately, it's a Latin dish (herba salata). The first recorded use in Ottoman Turkish is from 1533.

Thinking just of ingredients, every single seafood name I could think of is Greek (octopus, mussels, bluefin tuna/horse mackeral, shrimp, hamsi, seabass), with the only two exceptions: trout, which I believe is the main freshwater fish eaten in Turkey, and deniz börülcesi which is a type of seaweed. Both are Turkish words, though I would assume it was also eaten by the Greeks before the Turks reached the sea. Most names of fruits and vegetables come from Arabic or Persian, even for some relatively recent additions after the Ottomans were already more powerful than the Arabs or the Persians. The New World crop corn in Turkey is called mısır, which indicates where the Anatolian Ottomans got it via trade as it was originally called Mısır lahana, that is, Egyptian cabbage. Another good food name that shows trade relations is portakal, the modern Turkish for orange, comes from the Italian word Portogallo, meaning Portuguese. Meze, the quintessential Turkish culinary experience, comes from a Persian word meaning "taste, culinary delight"--however, this gets into the problem of using etymology for history, as it seems that the word existed in Turkish before it was applied to the practice of eating small plates, which seems to be a particularly Ottoman tradition that spread throughout the Empire (with the same name), from the Balkans to Arab World. Oh, and it's so good. It's so so good.

The last thing I'll say that these dishes don't necessarily reflect what most peasants typically ate even 50 years ago. I've been going through the notes of the anthropologist Paul Stirling who spent, several years in a Central Anatolian village, and while he barely mentions the foods, his wife Margaret was with him for much of 1949-1951. I'm lucky enough to have her notes as well (she had some anthropological training as well, though I believe only at the undergraduate level), and they're entirely different. She carefully notes what they are fed almost every day, and 1) rarely is there any meat (peasants worldwide in generally rarely have had the opportunity to meat), 2) there are a lot of things like porridge--things that have for the most part disappeared from Turkish menus, both out and at home, as far as I can tell.

Breakfasts might be something like "yogurt and bread and water for breakfast. bread and honey, kaysi [apricots], karpuz [watermelon]", " hard-boiled eggs and milk and bread", "bulgur boiled in milk, potatoes mashed, bread." Bulgur, maybe it's worth noting, is one of the few commonly used English words of Turkic origin, though the word ultimately comes from the Persian "barġul/burġul".

Lunches and dinners in Stirling's notes are typically along the lines of:

"Food - rice ([bought] from Kayseri [the local big city]), çorba [soup], grapes, bread."

"special - eggs, çorba [soup] (bulgar), a little halva [halva--of Arabic origin, but used in Turkish ~1300], hoşaf [fruit compote--Persian origin, but used in Turkish since ~1432] (uzum [grape]). [evening meal] cabbage stalked and chopped up with dried patlıcan [eggplant] [...next evening meal] potatoes and soğan [onion] mixed. One night he sent for halva [probably semolina helva, not the tahini halva more common in America] for me. Undoubtedly the leanest period for food."

"Plenty of milk - every house has at least a little [...] Eggs also plentiful. Fresh vegetables not yet in, - plentiful and cheap in Kayseri, not brought out. But 3 or 4 kinds of weeds gathered and eaten [probably including things like arugula, which is often collected wild and eaten in Turkey]"

These very simple dishes seem to be quite common. Only at weddings do we see more complex dishes:

"Meals at the düğün [wedding]. 'Yenges' [female in-laws] fed twice - once at girl's house, then at boy's, (i) at girl's house: 9 courses:- pirinç çorba [rice soup], mantı, baked apricots, baked figs, fried eggs or vegetables, sweet dish made from pekmez [grape molasses] and flour, plain rice pudding, stewed plums and üzüm [grapes] accompanied by a dish of plain bulgur, and finally village baklava. Loaves of bread accompanying meal. (ii) at boy's house, 6 courses - veg. soup, mantı, baked figs, rice pudding, dish of vegs., fruit stewed in pekmez - again with loaves of bread. As soon as bowls put on table, women tucked in and did not wait to chew food before putting more into their mouths."

From those notes, it's hard to identify the origins or even what dishes they were eating, but it'll suffice to say that peasant cuisine seems to be very plain and simple in this period, likely reflecting easily available ingredients more than anything else. Lots of very simple things like bread, eggs, fresh fruit, soups, cooked vegetables, and boiled grains such as bulgur (but not, to the Stirlings' minor annoyance, oatmeal. From Stirling's notes, "We mentioned eating yulaf [oats] and yulaf unu [oat flour]. They were surprised - thought it was only good for animal feeding. Oats not grown in this part, but they grow in the Adana region.").

Edit: what I forgot to emphasize is that when going through lists like this is finding out how many of elements of modern Turkish cuisine originate in Turkic Central Asia, before the Turks even got to Anatolia, considering how much basic elements of Turkish food just didn't exist in Central Asia--not just seafoods and agricultural crops that I don't imagine nomadic groups would use much of and that the Turks encountered in Anatolia, but also things that came over in the Columbia exchange like tomatoes. The Turkish version of "real women have curves", for instance, is "yemekte salça kadında kalça" (food has tomato paste and women have hips).

tl;dr: The roots a lot of Turkish cuisine predates both the Ottomans and the Turk's encounters with the Byzantines, but a lot of it is also borrowed from a variety of neighboring groups (Greeks, Arabs, Persians, Italians and more). There's no simple story of X cuisine is mainly based on Y, especially not in that region, because the relationships are way more tangled and multi-directional than that. The real trick you have to know is saying, "I don't know who made it first, but it's definitely best in [whatever country you're currently in]".