Add warm water, oil, and honey to flour mixture. Blend at low speed until moistened; beat 1 minute at medium speed. Switch to the dough hook and gradually mix in remaining flour to make a smooth dough, add more or less flour as necessary. Knead for 5 minutes.
Place dough in lightly oiled bowl and turn to grease top. Cover; let rise until dough doubles, about 45 minutes.
Preheat oven to 450° with a pizza stone on the center rack. (At least 30 minutes before baking pita so your stone gets really hot.)
Punch the dough down, and divide dough in to 6 pieces. Let pieces rest five minutes covered.
Form each piece of dough in to a ball, flatten and roll out each piece in to an eight inch circle. Keep the rolled-out pitas covered until ready to bake, but do not stack.
Bake pita two at a time on the baking stone until puffed and starting to brown, about 2 - 4 minutes. Turn pitas over and bake 1 - 2 minutes more. Wrap pita breads in a kitchen towel to keep warm until ready to serve.