Dissolve yeast in warm water in a large bowl; let stand 5 minutes. Lightly spoon whole wheat flour into dry measuring cups; level with a knife. Add 1 cup whole wheat flour to yeast mixture, stirring with a whisk. Cover and let rise in a warm place (85°), free from drafts, 2 hours (batter will be bubbly).
Stir in salt and oil. Lightly spoon bread flour into dry measuring cups; level with a knife. Add bread flour, stirring with a spoon. Turn dough out onto a lightly floured surface. Knead until smooth and elastic (about 10 minutes); add enough of remaining whole wheat flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel tacky). Coat the inside of a large zip-top plastic bag with cooking spray; add dough. Seal and refrigerate 12 hours or overnight.
Place pizza stone on the bottom rack in oven. Preheat oven to 500°.
Turn dough out onto lightly floured surface; shape into 12-inch log. Divide dough into 6 equal portions. Working with 1 portion at a time (cover remaining dough to keep from drying), shape each portion into a ball. Cover; let rest 1 hour.
Roll each ball into a 6-inch circle. Place 3 circles on pizza stone. Bake at 500° for 5 minutes. Transfer pitas to top rack; bake an additional 2 minutes. Cool on a wire rack. Repeat with remaining dough.