I'll see if i can find some links to endorse this
Particularly at its "greatest territorial extent." Its capital straddled the Bosporus, though it was mostly on the Western side which is Thrace and firmly in Europe. The Eastern side is on the extreme northwest corner of Anatolia - Asia Minor - also known as the Near East. Only very recently and then only sometimes conceptualized as belonging to the Middle East, even though it probably shouldn't be. The Ottomans had extensive territory in the Middle East including most of the coastline of the Arabian peninsula and Iraq, but then they had substantially more territory in North and East Africa and also in Europe - pressing as far into that continent as Vienna, completely overrunning the Balkans and encircling the entire Black Sea.
I know historically the Europeans thought of the Ottoman Empire as "the Orient," aka the East, but to me it seems very odd to characterize it as Middle Eastern.
A proper Middle Eastern Empire would be... the Babylonians, the Assyrians, the AlSauds, or the Persians. Some of which are what I thought of first. But then the clue didn't fit...