This is difficult to answere --for it pressupposes some  things that m,ay not be the case> that Turkish was indeed spoken in Greece by native populations. I think this was not really the case. It was probably a (pidginoid) lingua franca but no more than that. Of course, this is NOT to say that there was no impact. Siyeable vocabulary traces, no more or less than anywhere in the Balkans, remain in Greek. However, scarcely anything that one would call grammatical. Still, some grammatical interference may be noted in the Greek spoken in areas of Lesbos/Midilli. I hope this answers some of your questions. Cheers.

Thanks for the comment Costas. Rings a bell. I doubt as well that Turkish was spoken especially in the rural areas at all by native populations. On the other hand, indeed there are remarkable hints that a pidginoid form of it was used as a lingua franca. I personally know a folk song from Crete which is sung in a half Turkish half Greek langauge. Perhaps this is the language which they call Cretan / Giritçe I don't know.

However I am also interested very much in what happened during the Hunnic, Avaric, Oguric and Bulgaric periods, which all are almost sequential.  We know that many of these nomadic peoples had settled in the regions, which form the modernday Greece. There is no doupt that most of these peoples were Christinized and Hellenized later. But I wonder what was their initial social and linguistic impact on the society and the country. We know very well that Byzantine civilization lost great ground especially during the Avaric period. The lands down to the Pelopennese were "barbarized", destroying almost everything Byzantine.

Yes, the way you put it really is pretty interesting. I know of Girtice too --I have Turkish friends in Ayvalik whose grandmothers speak it; they, less than 50 y.o. today understand it. Good luck in your endeavours!