In the bowl of a stand mixer fitted with a dough hook, mix together the water, sugar and yeast. Let it sit until foamy, about 5 minutes.
With the mixer running on low speed, add the oil and salt, and then gradually add the bread and whole-wheat flours. Increase the speed to medium high and mix for 7 to 10 minutes, adding just enough additional bread flour so that the dough no longer sticks to the bowl. Do not add too much flour; the dough should be smooth and slightly sticky. Lightly coat a clean large bowl with oil or cooking spray and place the dough in the bowl, turning it once or twice to coat it in oil. Cover the bowl with plastic wrap and let it rise at room temperature until it has doubled in size, about 2 hours.
Turn the dough out onto a clean work surface and divide it into 12 equal pieces. Mold each piece into a ball by stretching the top and tucking the edges under. Place the balls 1 inch apart on a piece of parchment paper, cover them with plastic wrap and let rise for 30 minutes.
Preheat the oven to 500 degrees F and line 2 baking sheets with parchment.
Lightly dust a work surface with flour. With a rolling pin, roll out the balls of dough into 1/4-inch-thick circles. Place them on the lined baking sheets and bake until cooked through and puffy, about 5 minutes. You do not want them to brown. Cool on a rack and enjoy.
Leftovers can be frozen in an airtight container or ziptop bag and then reheated in the microwave or toaster.