Stretch and roll the dough on a lightly floured surface into a thin rustic round about 1/4 inch thick and 12-inches in diameter (no need to be perfectly round). Transfer to an 18-inch-long piece of heavy-duty aluminum foil and brush the pizza with the oil. Lift the whole piece of foil and put on the grill. Grill until the dough is charred on the bottom and almost cooked through, turning the foil a quarter turn a few times to ensure even cooking, 4 to 7 minutes.
Using the foil, remove the crust from the grill. Scatter the mozzarella and Parmesan over top, add the sliced tomato and season lightly with salt and pepper. Using the foil, lift the pizza back onto the grill. Cover and grill until the cheese is melted, 2 to 3 minutes more. Remove the pizza from the grill and top with the basil.
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