Preheat oven to 450 degrees, with racks in upper and lower thirds. Brush two rimmed baking sheets with oil, or for easy cleanup, line with parchment paper then brush with oil. Place one wrap on each sheet; brush with 1 teaspoon oil.
Sprinkle wraps with Asiago, then dollop with ricotta. Sprinkle with mushrooms and onion; season with salt and pepper.
Bake pizzas until crust is crisp and very brown all over, 20 to 25 minutes, rotating sheets from top to bottom and front to back twice during baking. Cut in half with a pizza cutter or knife; serve one half per person.