As Bill/SFNM says, high heat is very important. I make pita fairly often and have done so with different ovens in different places for many years. Whatever the oven’s highest heat is, that’s where I set it. I use a stone, but have also at times, as a guest baker at friends’ houses, used an inverted cast iron pan. Either way place the stone or cast iron pan as low as you can in the oven and preheat for at least an hour, longer if your oven is slow to come up to temperature.
The dough needs to be pretty thin when rolled or stretched out, and it really helps if the dough’s thickness is consistent all the way across and around the disc. (Typically I roll out to about 1/8” thickness. Sometimes thicker areas of dough won’t release for the puffing up and they can hold back the thinner areas of dough from puffing as well. I’ve always had the best results when using a rolling pin. When using the rolling pin, I try to use it in the same way one rolls out homemade pasta dough, a way that involves less pushing down on the dough than pressing lightly and pushing the dough forward in front of the pin and back when rolling it back. This stretches the dough rather than compressing it so much by just pressing down on it. And using the rolling pin helps make the thickness even which in turn facilitates the entire top layer popping up from the bottom like a balloon.
I’ve never used oil in pita dough, and I wonder if the oil might have an inhibiting effect on the rise.
Stephen
