Take pizza dough out of cooler at least 20 minutes ahead of time. Divide dough into quarters, shape each quarter into a ball, and then roll each ball into an 8-inch circle.
Brush each crust with oil and place oil side down on grilling grate over high heat. Brush top of crust with oil and carefully cover grill with foil. Grill until bottom is fully cooked, about 5 minutes, then turn over.
Quickly spread 2 tablespoons sauce over each crust, then divide the cheeses between them. Sprinkle with red pepper flakes to taste. Cover again with foil and grill until bottom is cooked and cheese is melted, about 5 more minutes.