"Cuisine" is the easy part since, as you note, we have recipes. "Culture" is harder, because we don't have much description of that. We have some descriptions of major feasts, with lists of food items like:

When Ashurnasirpal, king of Assyria, inaugurated the palace in Calah, a palace of joy, and erected with great ingenuity, he invited into it Ashur, the great lord and the gods of his entire country. He prepared a banquet of 1,000 fattened head of cattle, 1,000 calves, 10,000 stable sheep, 15, 000 lambs – for my lady Ishtar alone 200 head of cattle and 1,000 sihhu-sheep – 1,000 spring lambs, 500 stags, 500 gazelles, 1,000 ducks, 500 geese, 5000 kurku-geese, 1,000 mesuku-birds, 1,000 qaribu-birds, 10,000 doves, 10,000 sukanunu-doves, 10,000 other, assorted, small birds, 10,000 assorted fish, 10,000jerboa, 10,000 assorted eggs, 10,000 loaves of bread, 10,000 jars of beer, 10,000 skins with wine, 10,000 pointed bottom vessels with su’u-seeds in sesame oil, 10,000 small pots with sarhu-condiment, 1,000 wooden crates with vegetables, ...

but this isn't a typical meal - it's a 10 day feast for 69,574 people. Translation above from the version available quoted online at https://www.ancient.eu/article/730/the-banquet-stele-of-ashurnasirpal-ii/

The cuisine we know in part from archaeology and incidental references in texts. Grain (often as flat bread) was central, and was accompanied by various vegetables, fish, and meat (birds, lamb/mutton, beef, hunted wild meat), and beer and wine. Augmenting this, we have the recipes. Many of the recipes are stews, in which meat can play the central role. Other recipes include bird pie. Common flavourings are onions/leeks and garlic, cumin, coriander, and mint. Considering that onions/leeks, garlic, cumin, coriander and mint were still central to the flavouring of food in Medieval cuisine in the area, and are still used today, there appears to be a strong thread of continuity in Mesopotamian cuisine over the last 4,000 years.

To quote one of the shorter recipes (from Bottero 1987):

Goat-kid stew: Head, legs and tails will be (at first) singed over aflame. Meat is needed. You set water. You throw fat in it. Onion, samidu, garlic, blood, "sour(ed) milk (?)", you squeeze (them together in order to extract the juice which is to be added in the cooking pot). An equal amount, some plain suhutinnu (lit. "normal," isaru, that is neither prepared nor cooked, but added raw).

For a description of the Yale tablets with the recipes, see:

Bottéro, Jean. "The Culinary Tablets at Yale." Journal of the American Oriental Society 107, no. 1 (1987): 11-19. doi:10.2307/602948.

For the recipes in detail:

Jean Bottéro, Textes culinaires Mésopotamiens/Mesopotamian culinary texts, Eisenbrauns, 1995: https://books.google.com/books?id=sggKDtJMsLgC

For more on this, aimed at the general reader, with various useful background stuff:

Jean Bottéro, The Oldest Cuisine in the World: Cooking in Mesopotamia, University of Chicago Press, 2004: https://books.google.com/books?id=PxnaaTzC8tMC

For an account of cooking this cuisine: https://news.yale.edu/2018/06/14/what-did-ancient-babylonians-eat-yale-harvard-team-tested-their-recipes

For more on other sources of information, such as archaeology:

Cécile Michel, "L’alimentation au Proche-Orient ancien : les sources et leur exploitation", in B. Lion (ed), L’histoire de l’alimentation dans l’Antiquité: Bilan historiographique, Dialogues d’Histoire Ancienne Supplément 7, Besançon, 2012, p. 17-45. [in French]