Whisk water, 1 tablespoon oil, yeast and sugar in a large bowl; let stand until foamy, about 5 minutes.

Add 1 1/2 cups all-purpose flour, whole-wheat flour, cheese and salt to the yeast mixture; stir to combine. Add just enough of the remaining 1/2 cup all-purpose flour to make a smooth but slightly sticky dough. Turn out onto a lightly floured work surface and knead until smooth but still soft, about 5 minutes, adding more flour to the work surface as needed.

Clean and dry the bowl. Add the remaining 1 teaspoon oil to the bowl, add the dough and turn to coat with oil. Cover with plastic wrap and set in a warm place until doubled in size, 1 to 1 1/2 hours.

Punch down the dough and divide into 6 equal balls. Dust with flour.

Place a large skillet over medium-high heat until very hot. Roll or stretch each ball into a circle about 7 inches in diameter. Place one dough circle in the heated pan and cook, turning once, until browned on both sides and cooked through, 30 seconds to 1 minute per side. Transfer to a plate and repeat with the remaining circles of dough, adjusting the heat as necessary to prevent burning. Stack the cooked flatbreads on top of each other and cover with a clean towel; let cool completely before storing.

MAKE AHEAD TIP: Store in a plastic bag for up to 2 days.