In a small bowl, mix the honey with 1/2 cup lukewarm water and stir in the yeast. Let sit until the yeast has started to foam, 2 to 3 minutes.
Put both flours and the salt in the bowl of a stand mixer fitted with the dough hook. With the mixer running on low speed, mix in the olive oil until fully incorporated, about 2 minutes. Add the yeast mixture and 1 cup lukewarm water and mix until fully incorporated, 2 to 3 minutes more. Raise the speed to medium and knead the dough until it comes away from the sides of the bowl and forms a ball that is smooth, elastic, and no longer sticky, 4 to 5 minutes. If the dough seems too wet, add more all-purpose flour, 1 tsp. at a time. Raise the speed to medium high and continue to knead for 5 minutes. Oil a medium bowl, transfer the dough to the bowl, and roll to coat it with the oil. Cover with plastic wrap and let rise in a warm place until doubled in size, about 1 hour.
Turn the dough onto a lightly floured surface and knead two or three times to remove any air bubbles. Form into a log and divide it evenly into 10 pieces. Form each piece into a ball and then roll into disks about 1/4 inch thick and 6 inches in diameter. Lightly oil two rimmed baking sheets. Put the disks on the baking sheets, cover with plastic wrap, and let rise until just doubled in thickness, 30 to 45 minutes.
Prepare a medium-high gas grill fire, leaving one burner off, or prepare a charcoal fire with a hot zone and cool zone by pushing the coals to one side of the grill.
Brush a cast-iron griddle or skillet lightly with olive oil and put it on the grill over the cool zone. When very hot, cook the pitas in batches on the griddle until starting to brown, 2 to 3 minutes. Flip and continue cooking until lightly golden on the bottom and the pitas are cooked through, 2 to 3 minutes more. Cool on a wire rack. If stuffing, cut the pitas in half and carefully separate the layers with a fork.