I can't really speak to Byzantine armies, but I can tell you a fair bit about Ottoman armies.

First of all, it's important to remember that the vast majority of Ottoman armies weren't professional soldiers. They weren't even part-time soldiers. Most of them were mounted irregulars of some sort (usually classified as akinci). They served myriad purposes, but basically none of them were related to direct involvement in pitched battles. They were hit-and-run troops. You might see them involved in sporadic clashes with other cavalry, but for the most part that was the responsibility of the sipahis (both timariot and Kapikulu). The akinci were, in a sense, just like their steppe-bound predecessors. They harassed the enemy, provided scouting services, and they were paid pretty much exclusively out of loot -- it cost the Ottomans almost nothing to field them, so field them they did. If you see an Ottoman army of significant size prior to 1600, odds are half that force is made up of akinci and the related irregulars -- Crimean Tatars who were effectively "on loan" from the Crimean Khan, religious warriors (ghazis), sekban (mercenary) infantry that were hired on an ad hoc basis, azaps (the dismounted cousins of the akinci), and so forth. The Ottomans had a lot of irregulars.

Then there's the Timar system. Timariots, at least on a socioeconomic level, were essentially Ottoman knights. They were given land, almost always tax-free, in return for military service. The timariot himself was, depending on the wealth of his territory, responsible for bringing a number of armed retainers (cebeci) to battle. Wealthier tracts of land (ziamets and sanjaks) came with correspondingly higher expectations from the Ottoman state. Sanjaks are a little weird in that they weren't strictly in the timariot system but, for the purposes of the general levy, they were frequently counted as timars. This system, as of 1609, gave the Ottoman Empire a theoretical muster of about 200,000 men, including some ethnic levies (Turkoman and Kurdish, neither of whom were properly integrated into the timar system; they were often aghaluks, ruled by aghas [lords], rather than divided into fiefs -- these aghas weren't to be confused with the honorific given to officials, such as the Kizlar Agha, the man in charge of the eunuchs).

Then there were the professional formations -- Janissaries, Sipahis, and the artillery corps (Topcu). These were the Kapikulu ("slaves of the gate" -- a reference to the Sublime Porte, indicating direct service to the sultan). Contrary to popular belief, they rarely marched to war en masse, especially if the sultan didn't go. Only a handful of Janissary ortas, for instance, would've been present at battles like the Siege of Malta.

These staggering numbers that the Ottomans could muster were, as you can probably imagine, difficult to feed. The Ottomans solved this issue in four ways. First, the Ottomans had imperial granaries located along the major roadways they'd use in wars every three leagues. This meant that armies could resupply while on the move. Local officials would be responsible for ensuring the granaries were sufficient. Second, the Ottomans foraged. A lot. The Ottoman campaign season was dictated as much by the availability of forage (specially for their apparently endless supply of horses) as weather. Third, the Ottomans used ships and wagons to resupply on a regular basis. Finally, the Ottomans campaigned with field-portable ovens, so the troops could cook their own bread in the field. Decentralizing bread production like that meant individual ortas were surprisingly self-sufficient. And if units are self-sufficient, that means they can take different routes and arrive at the same location. This allowed the army to spread out while in friendly territory, like Rumelia and Anatolia, and similarly spread out the burden of feeding all these troops, making it easier for armies to march by and not starve towns to death.

Incidentally, the Ottomans had the hardest time by far when they couldn't forage. The Persians in particular proved troublesome opponents because of their ability to effectively hem the Ottomans in -- they were one of the few rivals that could reliably contest Ottoman raiders in the field.