Step 1

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.

Step 2

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.

Step 3

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.)

Step 4

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.

Step 5

Place a baking sheet, cast-iron pans or a pizza stone on the middle rack of the oven and preheat the oven to 450°F.

Step 6

Once the oven has heated, arrange a few of the rolled-out disks onto the hot surface, at least 2 inches apart. Do this by placing a disk on the palm of your hand and flipping your hand to drop the disk onto the surface, being careful not to burn yourself. The pitas will puff up almost immediately. Bake for 2 minutes, flip over with tongs and continue to bake for 1 more minute. Remove the pitas from the baking surface and place them in a towel-lined basket or bowl. Fold the towel over the pitas to keep them warm. Repeat with the remaining dough disks.

Step 7

Heat a medium-size skillet over medium-high heat and brush on a very thin layer of oil. You'll know the pan is hot enough to cook the pitas when it sizzles after you've sprinkled on a few drops of water.

Step 8

Place a disk in the hot pan. Cook until you see bubbles begin to form, about 2 minutes. Flip the disk over and cook for 1 minute. It should begin to puff up a bit. Flip over again and cook for 1 more minute on the other side. Transfer to a towel-lined basket or bowl to keep warm. Continue to cook the remaining pitas, brushing a very thin layer of oil onto the pan in between cooking. Place the pitas in a towel-lined basket or bowl. Fold the towel over the pitas to keep them warm. Repeat with the remaining dough disks.