In the bowl of a stand mixer, use a wooden spoon to mix 1 cup of the all-purpose flour, ½ cup of the whole-wheat flour, 1½ cups warm water, the yeast, and ½ tablespoon of sugar. Rest, uncovered, until puffed and foamy, 20 to 25 minutes.

Add the rest of the all-purpose and whole-wheat flours, the remaining ½ cup water, the remaining 1 tablespoon sugar, the olive oil, and the salt. Attach the dough hook and bring the dough together over low speed, then raise the speed to medium and knead the dough until slightly springy but still slightly soft and sticky and pulling away from the side of the bowl, 5 to 6 minutes (add flour by the tablespoonful if needed, but try not to add too much). Remove the dough hook, cover the dough with a clean kitchen towel, and let it rest in a warm place until doubled in size, 45 minutes to 1 hour. Refrigerate the covered bowl for another 45 minutes to 1 hour (this helps develop the flavor and texture of the dough).

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Flour a work surface, uncover the dough, gently transfer it to the board, and shape it into a rectangle about 1 inch thick. Use a sharp knife to cut the dough into 8 equal pieces, then form each piece of dough into a ball, tucking the ends underneath and pinching them together gently to help encourage a nice round shape. Cover the dough balls with a clean kitchen towel and let them rise for 30 minutes.

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While the dough is rising, take a clean cast-iron skillet or griddle and scrub off any bits stuck to it (these can burn). Arrange a rack about 8 inches from the broiler, invert the skillet, and place it on the rack. Preheat the oven to the highest it will go (500° or 525°, or even 550°F).

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Uncover the dough, gently flour each ball, and use a rolling pin to roll the dough into a circle by rolling up and down to form a 6-inch oval. Rotate the oval 90 degrees, then roll the dough again; it should roll into a beautiful 5-inch round. As you roll the dough into rounds, place them on the towel and cover them with another clean towel and let them sit for 10 to 15 minutes, until the pitas have fluffed up to about twice their original height.

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Uncover the dough, open the oven, and place one of the rounds flat on the skillet, working quickly and closing the oven so it holds its high temperature. Bake the pita until it’s puffed and golden, 2 to 2½ minutes. Using a spatula, remove the pita from the skillet and repeat with the remaining pitas, letting the oven and skillet preheat for a minute or two between pitas (if you have a really large skillet or a long cast-iron griddle, you can do two pitas at a time). Let the pitas cool, then ideally eat while still warm. Once cooled, wrap individual pitas tightly in plastic wrap and freeze for up to 3 months. To serve, defrost, then toast in a toaster oven; heat in a 400°F oven for 3 to 4 minutes; or warm directly over a flame on your stovetop.