Greek(?) flatbread

I used to frequent a souvlaki stand near Fraunces Tavern in New York. The proprietor, a charming old man who was born and raised in Greece, used a flatbread that he called "pita" but was unlike most pita bread. It had a slightly glutinous texture and no pocket, so it couldn't be split and filled. He just wrapped it around the meat and pulled out the skewer.

Perhaps it had a different name in Greece and he only called it pita because that's what his customers understood. If only I had asked him to write it down for me! I've found pocketless pitas in grocery stores, but they have the usual bready texture.

Any idea what this Greek flatbread is, and how to make it?

Janet