The Double-Headed Eagle was the emblem of the Byzantine Empire.
The double-headed eagle was the symbol of the Paleologoi, the last Greek-speaking “Roman” (i.e. Byzantine) dynasty to rule from Constantinople. The Emperor Michael VIII Paleologos recaptured Constantinople from the Crusaders in 1261, from a state based in Asia Minor; the double-headed eagle symbolized the dynasty’s interests in both Asia and Europe.
Most obvious of these is the Greek Orthodox Church, centred in theory in Istanbul to this day, and so it is not surprising that the Church would use the flag.
Source: Newsfinder.org