1 (1/4-ounce) package active dry yeast (2 1/2 teaspoons)
1 teaspoon honey
1 1/4 cups warm water (105–115°F)
2 cups bread flour or high-gluten flour, plus additional for kneading
1 cup whole-wheat flour
1/4 cup extra-virgin olive oil
1 teaspoon salt
Stir together yeast, honey, and 1/2 cup warm water in a large bowl, then let stand until foamy, about 5 minutes. (If mixture doesn't foam, discard and start over with new yeast.)
Whisk 1/2 cup flour (either whole wheat or bread flour, doesn't really matter, just remember to adjust the total in the final addition of flour) into yeast mixture until smooth, then cover with plastic wrap and let stand in a draft-free place at warm room temperature until doubled in bulk and bubbly, about 45 minutes. You can let this sit for several hours with no ill effects. Really, I let mine go from breakfast until lunch, when I can get back home to add the rest of the ingredients.
Stir in oil, salt, remaining 3/4 cup warm water, and remaining 2 1/2 cups flour mixture until a dough forms.
Turn out dough onto a floured surface and knead, working in just enough additional flour to keep dough from sticking, until dough is smooth and elastic, 8 to 10 minutes. Form dough into a ball and put in an oiled large bowl, turning to coat. Cover bowl with plastic wrap and let dough rise in draft-free place at warm room temperature until doubled in bulk, about 1 hour (again this can sit for several hours without any issues).
Punch down dough and cut 2 pieces. Put one ball in plastic bag in refrigerator (for making pizza with later in the week). Cut remaining piece in to 4 pieces and roll into balls. Flatten 1 ball, then roll out into a 6 1/2- to 7-inch round on floured surface with a floured rolling pin. Doesn't have to be perfect, just make them pretty thin. Transfer round to 1 of 2 baking sheets lightly sprinkled with flour or cornmeal. Make 3 more rounds in same manner, arranging them on baking sheets. Loosely cover pitas with clean kitchen towel (not terry cloth) and let stand at room temperature 30 minutes. I do all of this on one of those perforated pizza pans, and then just put the pan in the oven, but the rounds do transfer easily, so don't worry about it.
Set oven rack in lower third of oven and remove other racks. Preheat oven to 500°F.
Transfer 4 pitas, 1 at a time, directly onto oven rack. Bake until just puffed and pale golden, about 2 minutes. Turn over with tongs and bake 1 minute more. Cool pitas on a cooling rack 2 minutes, then stack and wrap loosely in a kitchen towel to keep pitas warm.