This brief discussion on Byzantine food, far from in-depth, is designed to give some background to Greek and Turkish food.
Byzantium, later named Constantinopole, was a city on a peninsula, surrounded by water on three sides. The citizens called themselves Romans, not Greeks.
The bleak years of the Middle Ages only happened in Europe; wealth, luxury, learning and knowledge continued on in Constantinopole. It was the Byzantine doctor Anthimus who wrote the last cookbook to come out of the Western Roman Empire, shortly after 500 AD.
Early Byzantium was a Greek fishing colony, exporting fish and seafood. It was founded in 667 BC by settlers from Megara, Greece (a city still extant, 21 miles / 35 km west of Athens), led by a man named Byzas, whom Greek legend says was directed by the oracle at Delphi to settle there. In the Roman Empire, it became known its Latin name of Byzantium.
By 324 , Constantine I, Roman Emperor (born 272 AD, reigned 306–337 AD), designated Byzantium as the second capital of the Roman Empire, to look after the eastern half of the empire. It was renamed to Constantinopolis (polis meaning "city" in Greek) in 330 AD. The Roman Empire was divided into east and west in 395 AD, with Byzantium the eastern capital.
When the western half of the Roman Empire fell (many use the arbitrary year of 476) Byzantium became the new capital of the Roman Empire. Taking into account the founding of the Empire in Rome, and its continuation on in Byzantium, the Roman Empire actually played a role in history from 510 BC to 1453 AD -- nearly 2000 years.
92 Emperors reigned at Byzantium, of which the last was Constantine XI, and over 70 nationalities were in the Byzantine Empire. Balat and Hosoy were Jewish areas of the city of Byzantium; they lived completely equal, and even had their own law courts.
Byzantium was conquered by Muslim Turks on 29 May 1453 in the afternoon, led by Mehmed II, aged 20 at the time.
Parts of the city surrendered rather than fight to the end, such as the present day Koca Mustafa Pasha area in the south-west of the city (then called "Psamatya.") Because of this, Psamatya was not only spared pillage, but Mehmed II insisted that the churches there not be damaged.
Parts of the Empire, such as the Galata suburb across the horn from Constantinople, were given so much religious and cultural freedom under the Muslims that they even continued to hold carnivals before Lent every year.
The Turks referred to themselves as "Ottomans", and saw the term Turk only as a pejorative term for peasants in Anatolia.
The city was renamed from Constantinople to Istanbul in 1930. Many Greeks still refuse to recognize the new name.
FOODByzantium became the bridge that carried the food of the ancient world into modern-day Greece and Turkey. It never experienced the Western European Dark Ages (roughly from 500 to 1300 AD)  -- though in our Western European centric history, we aren't taught that. During that time Byzantium was an ongoing, though of course evolving, culture and society.
There are not, though, many Byzantine cookbooks. What we know is from writings by travellers, church rules, laws, doctors writing about diets, etc.
The two main influences on their food were the Church, and the "science" of dietetics -- the theory of four humours in the body which could be balanced by what you ate for good health.
There were three meals a day -- breakfast, lunch and dinner. For lunch and dinner, wealthy people would have a starter, a main course, and a dessert.
The Byzantines had eggplant, oranges and lemons. They expanded our range of jams, jellies and fruit preserves. They made rice pudding, served with lashes of honey.
The poor ate a lot of salt pork and cabbage.
There was much street food. The most common cooking method was boiling.
The Byzantines made baklava, and called it "kopton." There is a recipe for it in Athenaeus, XIV, 647-48.
The Byzantines invented marzipan, the samovar for tea, and it is presumed, the fork. The number of tines in their forks increased over time. By the 1100s, Byzantine forks had five tines. Venetians brought the fork back to Italy.
They used simple sauces made with vinegar and oil, or vinegar and honey.
They used capers, dried figs and walnuts as starters for meals.
Mizithra cheese was made in Thessaly and Macedonia, and prosphatos cheese made in Crete.
BreadBakers in Byzantium were exempt from being called up for any public service, including military, in order to keep the bread ovens rolling.
Paximadion was a Byzantine bread made of barley, baked, then sliced, then baked again until very dry and hard. It was used for travellers and given out as army rations. The bread was named after the Greek cook Paximus. Today these are called "paximadi" in Greece, "beksemad" in Turkey.
Trakhanas was bread made from cracked grains such as emmer, mixed with soured milk, then formed into balls and dried.
Boukellaton were dried loaves of ring-shaped bread.
Piston was a porridge made from millet.
Meat & FishMost houses kept pigs. Lamb was more expensive. Beef was only seldom eaten. Cattle were used to pull ploughs instead, and were seen as work animals.
Herds of gazelles and wild donkeys were kept to be hunted as game.
The Byzantines loved botargo (salted mullet roe, aka "Bottarga" now.) By the 1100s, they were making caviar.
No meat or fish were allowed on Orthodox Church fast days.
FlavouringsByzantines had all the spices the Romans did, plus nutmeg and ambergris. Among flavourings used were anise, caraway, cinnamon, cumin, honey, mastic, pepper, salt, and vinegar. Reputedly, they were the first to apply the herb rosemary to lamb, and the first to use saffron. This made them the first to use rosemary and saffron in non-medicinal applications.
They used mastic a lot as a spice.
Ordinary people, of course, wouldn't have had access to the wide variety of spices that the rich did. But even the poor seem to have had access to some spices, though not much meat.
Byzantines used garum as did the Romans and earlier Greeks. They called it "garos."
The Byzantines mixed sweet and savoury. They used caraway in savoury dishes.
WineByzantines were fond of spiced wines. They particularly made wine flavoured with anise, called "anisaton", which later evolved into ouzo.
They also made wines flavoured with absinthe, aniseed, chamomile, gentian, ginger grass, mastic, putchuk, rose, spignel, spikenard, stone parsley, storax, tejpat, valerian, violet, and yellow flag, approaching the complexity of today's vermouths.
They put resin in wine (like today's Greek retsina.) Resined wine was unpopular even back then amongst Westerners. An ambassador to the Byzantine Court, a Lombard named Liutprand of Cremona (C. 920-C. 972) wrote: "To add to our calamity the Greek wine, on account of being mixed with pitch, resin, and plaster was to us undrinkable " 
A common drink for soldiers was "phouska" -- the Greek name for the same drink Roman soldiers had had, "posca."
 Dates quoted for when Byzantium was chosen an offical city vary from 324, to 330, to 335 AD. Centénaire de la société nationale des antiquaires de France, Paris, 1904, p. 281 says 324. Official inauguration of new name was in 330 AD.
 Opinions vary wildly when to date the end of the European Dark Ages, and whether in fact Dark Periods would be a more accurate term. That is an issue for students of other topics to discuss.
 Liutprand of Cremona: Relatio de Legatione Constantinopolitana (Report of his Mission to Constantinople) from Henderson, Ernest F. Select Historical Documents of the Middle Ages . London: George Bell. 1910, pp. 440-477.
Literature & Lore
"Get [your tuna] from Byzantium, if you want it to be good. . . ." -- Archestratus, fragment 62. Circa 350 BC. Olson and Sens translation.
"And therefore I have sailed the seas and come To the holy city of Byzantium." -- William Butler Yeats (13 June 1865 – 28 January 1939)