Mixer Method: Add the white flour, wheat flour, kosher salt, and instant yeast to the bowl of a standing mixer. Mix everything together and then add the water and olive oil. With the dough hook, mix on low until the dough begins to come together. Then increase the speed to medium and knead the dough for 8-10 minutes. The dough should be very soft and smooth and just a little tacky to the touch. As you are kneading the dough, look at it and add more flour or water (just a bit at a time!) if the dough looks or feels too dry or too wet.
Hand Method: Reserve a few tablespoons of white flour, then combine all the rest of the ingredients in a bowl until you have a shaggy dough. Sprinkle a board with the reserved flour, turn out the dough onto the board, and knead the dough until it is smooth and elastic and a little tacky but not sticky.
Shape the dough into a ball and oil it. Place the dough into a lightly oiled mixing bowl. Cover the bowl with plastic wrap, and let the dough rise in a warm place until the dough doubles, about 1 1/2 hours.
Meanwhile, about 1 hour before you are going to bake the pitas, begin heating your oven. Adjust an oven shelf to the lowest level and put a heavy baking sheet, cast iron pan, or baking stone on it. Preheat the oven to 475F.
Once the dough has doubled, use a bench scraper to divide the dough into 8 equal pieces. Shape each of the pieces into a ball by pulling the dough down and pinching it at the bottom. Cover the dough with a damp cloth at let it rest for 10 minutes.
Working with one ball of dough at time (keep the rest covered with the cloth), roll each out into a 4-inch disk. Let the dough rest another 10 minutes to allow the gluten to relax again. Then roll each piece of dough out into a 7-8 inch circle. If it's not working set that piece aside and roll out the rest and then come back to it and give it another roll.
When you are ready to bake each pita, turn it over and then brush or spray the pita with water (only the new top side - leave the side which will be on the hot stone/pan dry). Quickly place the pita onto your heated surface and bake it for 2-3 minutes.
As your pitas finish baking, transfer them to a plate and cover them with a towel. Let the pitas cool under the towel.
Heat a cast iron skillet (or a heavy skillet) over medium-high heat. Lightly oil a pita and brown the pita on both sides. This will take 30 seconds to 1 minute.