Coat a medium bowl with 1 tsp oil, then set aside.
In the bowl of a KitchenAid mixer, attach the dough hook and then add the 1 1/4 cup Flour, 1 1/4 cup Whole Wheat Flour, 2 1/4 tsp Yeast and 2 tsp Sugar.
Whisk the dry ingredients until combined, around 5 seconds.
Measure the water in a microwave safe cup and then add it to the dry ingredients in the KitchenAid mixer bowl.
To the dry ingredients, measure and add the 1/4 cup Plain Greek Yogurt and 2 Tbsp Oil and mix until a smooth ball forms, around 3 minutes.
If the dough is dry, add 1 tsp water at a time and mix after adding each tsp, with a total of 2 Tbsp water, until the dough is slightly sticky.
Add the 2 1/2 tsp salt to the dough and knead on low for 10 minutes.
Transfer the dough into the prepared bowl and then flip it over to coat the entire ball with oil.
Cover the bowl with plastic wrap and rest until risen, around 1 to 1 1/2 hours.
Dust a rimmed baking sheet with flour.
Transfer the dough to the counter and then using a bench scraper, divide the dough into 10 pieces, around 2 ounces each.
Form dough ball into a tight ball and then place them on the baking sheet.
Brush each ball with 1/2 tsp of the remaining oil, then cover them with a damp towel and rise for 30-60 minutes.
Preheat the oven to 500 degrees Fahrenheit with a baking stone in the upper-middle rack.
Form the Pita Breads
Dust two rimmed baking sheets with flour and then lightly dust the countertop.
Place a Pita Bread Dough ball on the counter, then using a floured rolling pin, roll the dough into a round 1/8-inch thick and 5 1/2 in diameter.
Transfer the dough round onto the prepared baking sheet, in a single layer.
Repeat to make each dough round.
Cover with a clean kitchen towel and let rest for 10 minutes.
Bake the Pita Bread
If you have, use a peel that is slightly dusted, to transfer two dough rounds to the preheated oven on the baking stone. Try not to flip the dough.
Immediately close the door once the dough is on the baking stone.
Bake until the Pita Breads have puffed and are lightly browned, around 3 minutes.
Use the peel to remove the Pita Bread from the oven and transfer to a wire rack and cover with a dry kitchen towel.
Repeat with each of the dough rounds until they are all baked.