Make dough: In a large bowl, whisk together 1 cup all-purpose flour, yeast, honey, and 1 cup warm water until smooth. Cover with plastic wrap; let rise in a warm spot until doubled in bulk, about 30 minutes. Stir in remaining 3 ½ cups all-purpose flour, whole-wheat flour, salt, oil, and remaining 1 ¼ cups warm water.

Knead dough: Transfer dough to a floured surface. Knead dough, dusting hands and surface with more flour as needed, until smooth and elastic, 10 minutes. Transfer to a large oiled bowl, turning to coat. Cover and let rise again until doubled in bulk, about 45 minutes.

Form dough into ball: Punch down dough, and form into a ball; then turn out onto a floured surface.

Divide dough: Quarter dough. Working with one piece at a time (drape a kitchen towel over the rest), divide each quarter into 4 smaller pieces.

Roll each piece, pinch, and flatten: Roll each piece into a ball and pinch, tightening the ball. Turn pinched-side down and flatten with your palm.

Use rolling pin to form rounds: Flatten each ball into a 6-inch round with a lightly floured rolling pin.

Let rest: Transfer rounds to rimmed baking sheets sprinkled with cornmeal; drape with kitchen towels. Let rest 30 minutes.

Bake: Meanwhile, preheat oven to 500°F and set an inverted rimmed baking sheet on rack in lowest position. Place 4 dough rounds on preheated sheet. Bake until puffed, 2 minutes. Flip and bake until golden in spots and just cooked through, 1 minute more.

Transfer to towel-lined basket: Transfer to a basket lined with a kitchen towel; cover to steam and keep warm. Bake remaining pitas and serve.