Oil the grill grates and decrease the heat to medium. Brush the dough with 1 to 2 teaspoons of olive oil and flip onto 1 end of the hot grill, leaving room for the tomatoes on the grate. Put the prepared tomatoes on the grill, close the lid and cook until the bottom of the crust is golden brown and the tomatoes are softened, about 1 to 2 minutes.
Brush the raw side of the dough with 1 to 2 teaspoons of olive oil, then immediately flip using the peel. Top with the grilled tomatoes, smashing and spreading the tomatoes to create a sauce. Sprinkle with the Parmesan, mozzarella and basil. Close the lid and cook until the bottom of crust is golden brown and the cheese has melted, another 1 to 2 minutes.