Make the dough: In a liquid measure, stir the sugar into 1 cup of lukewarm water. Stir in the yeast and set aside until the yeast is foamy, 5 to 10 minutes.
In the bowl of a stand mixer, stir together both flours and the salt. Make a well in the center and pour the yeast mixture, 1/3 cup of the oil, and 1 cup lukewarm water into the well. Mix with the dough hook on low speed until the dough becomes smooth and elastic and gathers around the hook, 4 to 5 minutes.
By hand, shape the dough into a ball. Wipe out the mixing bowl and put the dough back in the bowl. Drizzle the dough with the remaining 2 Tbs. oil and turn the dough to coat lightly all over.
Cover with a cloth and set aside in a warm place to rise until doubled in size, about 1 hour.
Gently deflate the dough with your hands, cover, and let it rest for 20 minutes.
Shape the pitas: Turn the dough out onto a lightly floured counter. Divide the dough into 12 even pieces, about 3-3/4 oz. each.
Shape each piece into a rough ball and then put each ball on an unfloured section of counter, cup your hand over it, and quickly rotate your hand over the dough. As long as the dough is a little stuck to the counter, this motion shapes the dough into a tight, evenly round ball.
On the floured part of the counter, roll each piece into a 1/8-inch-thick round that's about 7 inches across. As you finish each round, set it aside on a lightly floured surface. When all the dough is rolled, cover the rounds with a damp cloth (or two) and let them rest again for about 1 hour—they'll get a little puffy but won't double in size.
Meanwhile, position a rack in the bottom of the oven and heat the oven to 500°F.
Bake the pitas: Lightly sprinkle the dough rounds with salt. Arrange as many of the rounds as will fit without overlapping on an unrimmed, ungreased baking sheet and bake until the pitas begin to turn golden on top, 5 to 6 minutes. As each batch comes out of the oven, stack the pitas 3 or 4 high and wrap in clean kitchen towels.
Serve immediately or let cool to room temperature. Well wrapped, they’ll keep for 3 days in the refrigerator or 6 months in the freezer. Reheat in a warm oven to soften before serving.