Combine the yeast, warm water and honey in a bowl. Set aside for 10 - 15 minutes until foamy.
Combine the flours and salt in a large bowl of a standing mixer fitted with a dough hook. Pour in the water/yeast mixture and run the mixer on the second speed (second from the lowest speed). As the flours absorb the liquid, increase the speed to mid-range.
Once the liquid is completely absorbed, add the olive oil and continue to run with the dough hook until the incorporated. Dump the dough out onto a lightly floured work space. Add some of the additional 1/4 cup whole wheat flour if the dough is too sticky to knead. Cover with plastic or an up-ended bowl and rest for 2 minutes.
Knead the dough, using a dough scraper if needed, for 2 minutes. Rest the dough for another 2 minutes under a bowl or sheet of plastic. Repeat this cycle 4 or 5 times until the dough becomes smooth and pliable.
Splash a little olive oil in a bowl and turn the dough to coat. Loosely cover the bowl with plastic or a tea towel and set aside in a warm, draft-free place until doubled in size, about an hour. See Cooks Notes in the post for tips on this step.
Pre-heat oven to 475˚F. Place a pizza stone on the bottom rack or dust a large baking tray with flour. I heated the oven for one hour before baking. Lightly beat an egg with a small splash of water.
Punch down the dough, turn out onto a lightly floured work space, and divide into 8 pieces. Roll each piece into a ball and flatten into a circle. One at time, while keeping the remaining disks covered with plastic, use a rolling pin to gently flatten each piece into an 8-inch circle.
Brush well with the egg wash and using your fingers, lightly press a border around the edge of the dough disk about 1/2" from the edge. You can also cup your hand in a "C" shape and press down on the edges.
Dip your fingers into the egg wash and with a claw hand, drag 3 fingers across the dough to form rows. Repeat, to create a total of 6 rows. Be careful not to break through the dough. Sprinkle with black and white sesame seeds.
Place two rounds of dough onto a floured wooden paddle, and cover the remaining dough loosely with plastic. If you don't have a paddle, turn a baking sheet upside down, place a sheet of parchment paper on top, and place the pita rounds on top of the parchment paper. Slide the dough rounds onto the hot pizza stone. If using an upside-down baking sheet, the parchment paper will slide off the sheet onto the baking stone.
Bake for 6 minutes until lightly browned. They may puff up in the oven from trapped hot air inside the dough. They bake very quickly, and if baked much longer will become crisp like a cracker as they cool. Don't worry if they don't puff up - it won't make any difference to the flavor.
Repeat with the rest of the pita rounds, baking two at a time, until they're all baked.