Model your dough after the thin, crispy crust renowned chef Todd English serves at his Figs restaurants.
In the bowl of an electric mixer fitted with the dough-hook attachment, combine the whole-wheat flour, all-purpose flour, yeast, salt, and sugar. With the mixer on low speed, gradually add the oil and 1 1/3 cups lukewarm water. Knead on low speed until the dough is firm and smooth, about 10 minutes.
Divide the dough into four balls, each about 7 1/2 ounces. Line two baking sheets with parchment paper. Place two balls on each sheet, and cover with a damp towel. Allow the dough balls to rise in a warm spot until they have doubled in size, about 2 hours.
To roll out the dough, cover your fingers with flour, then place a ball on a generously floured work surface. Press down on the center of the ball with the tips of your fingers, simultaneously stretching the dough out with your hands. When the dough has doubled in width, use a floured rolling pin to roll the dough out until it is very thin. The outer border should be slightly thicker than the center. Transfer the dough on a spatula (allowing the dough to fold up almost like an umbrella) to a pizza peel or another flat, wide surface. The dough will be more oval than round. If any holes have appeared, pinch them closed. Repeat with the remaining balls.