Mix water, yeast, sugar and oil in a small bowl; set aside. Whisk together the flours and salt in a large bowl. Pour in the yeast mixture and mix together with wooden spoon until dough forms. If using a stand mixer, use the dough hook attachment to knead at medium speed for 10 minutes. Alternatively you can knead by hand for 10 minutes, until dough is smooth and elastic.
Transfer the dough to an oiled bowl, turning to cover with oil. Cover bowl tightly with plastic wrap or kitchen towel and let rise in warm place until doubled, about 1 hour.
Punch down dough and transfer to clean work surface lightly dusted with flour. Form 8 to 9 dough balls, cover with clean damp towel or plastic wrap and let rest for 20 minutes.
Working with one dough ball at a time on a clean and lightly floured work surface, use rolling pin to shape into about a 6-inch disc (or just smaller). Do not roll too thin. Place on a greased or lined baking sheet, cover with clean kitchen towel and let rest at room temperature for 10 minutes.
Preheat oven and baking stone to 450 degrees F. Bake 2 to 3 pitas at a time for 5 minutes. They should puff at around 3 minutes. When removing pitas from oven, do not remove the stone so to keep the stone hot. Wait 2 minutes before baking the next batch, to allow the stone to reheat to the correct temperature.
The pitas are best the same day they are baked. To preserve their freshness for another day, once they are cooled to room temperature, transfer to a freezer bag and store in the freezer for up to 2 months. Defrost in the microwave for about 20-30 seconds.
Notes
These pitas puff up beautifully so that when you slice into them they have the perfect pocket you need for your favorite gyro. You can alter the variation of flour used (e.g., use only all-purpose flour, or all whole wheat, or another combination) but my favorite is listed above.