Mix the mozzarella cheese and Parmesan together in a medium bowl, then set aside. Roll out the dough on a lightly floured work surface to 1/4-inch thick. Using a 2 3/4-inch-diameter cookie cutter, cut out dough rounds.
Add enough oil to a heavy large frying pan with 3-inch sides to come halfway up the sides of the pan. Heat the oil over medium heat until hot. Drop a scrap of dough into the oil to ensure the oil is hot enough. If the dough sizzles, then the oil is hot enough. Working in batches, add the dough rounds to the hot oil (being sure not to crowd them in the pan) and cook until golden brown, about 3 minutes per side. Using a slotted spoon, transfer the rounds to paper towels to drain the excess oil. Immediately sprinkle the cheese mixture over the hot pizzas. Serve warm.