Mix together 1 cup warm water, the yeast, and the sugar in a small bowl and let stand until foamy, about 5 minutes.
Combine the bread flour, whole wheat flour, and salt in the bowl of a stand mixer fitted with the dough hook or a large mixing bowl. Mix using the stand mixer or a wooden spoon until blended. Add the yeast mixture and the oil and mix until the dough forms a ball that pulls clear of the sides and bottom of the bowl, adding additional water a tablespoon at a time, if necessary. The dough should feel tacky when slapped with a clean hand, but it should not stick. If it sticks, add more flour, a tablespoon at a time.
Cover the dough with plastic wrap and let rise at room temperature until doubled in size, about an hour. Alternatively, let it rise in the refrigerator overnight.
Preheat the oven to 475°F. Place a baking stone or an inverted baking sheet in the oven to preheat as well.
Divide the dough into 8 pieces and roll into balls the size of baseballs. Cover with a damp cloth and let rise until they are about the size of softballs.
Roll each dough ball to about 1/4″ thickness with a floured rolling pin on a floured work surface. Cover with a damp cloth and let them sit for about 5 minutes. Flip one or two at a time onto the baking stone or baking sheet (so that the side that was previously facing up when you rolled the dough out is now facing down) and bake until puffed and cooked through, about 3 minutes. Remove with tongs and cover with a cloth to keep warm. Repeat with remaining pita rounds until all baked. Serve warm, or store in a sealable plastic bag.