Combine the warm water, sugar, salt, olive oil and yeast in a large bowl. Stir in the sourdough starter discard. Add half of the all-purpose flour and all of the whole-wheat flour and stir until well combined. Add the remaining all-purpose flour, half a cup at a time, mixing well after each addition.
Form the dough into a ball. If the dough is too sticky to work with, add 1 to 2 tablespoons of flour or more as needed. Turn the dough out onto a floured surface and knead for about 5 minutes, until smooth and elastic. Wipe out the bowl and oil it lightly. Place the dough in it and turn it over to lightly coat the entire ball. Cover the bowl with a plate. Let the dough rest for 1 hour in a warm spot, until doubled in size.
Punch down the dough. If you won't bake the pitas immediately, store the dough in the refrigerator for up to 5 days. Because the dough may double in size in the refrigerator, choose an appropriately sized container. (When you want to bake, remove as much dough as you’d like.)
Turn the dough out onto a floured surface and cut it into 12 pieces. Roll into balls between your palms. With a rolling pin, roll the balls into 7-inch disks, between ⅛- and ¼-inch thick. As you work, add flour to the work surface as necessary to ensure that the dough doesn't stick.