Pizza on an outdoor grill is a great alternative to oven-cooked pizza. If you love crispy crust, this is for you. Note: cook time is relatively short, so alternative toppings like uncooked sausage or other meats may not cook fully.
Preheat an outdoor grill for high heat, and lightly oil the grate.
Place pizza dough in a bowl to rise 2 to 3 times the original size, about 1 hour. Flour a work surface and a rolling pin. Divide dough in half; roll each half to a 10- to 12-inch diameter circle.
Place pizza dough rounds on the preheated grill using a wooden paddle (pizza peel) and close lid; cook until grill marks appear on the bottom of each round, 3 to 5 minutes. Remove from grill and flip over on a work surface. Lower grill heat to medium.
Brush each round with olive oil. Spread half the Asiago cheese on each crust; add tomatoes, basil, and mozzarella. Season with salt and pepper.
Transfer pizzas to grill with wooden paddle and close lid. Cook until bottom of rounds is cooked with grill marks and cheese is melted, 7 to 10 minutes. Remove from grill and let stand for 5 minutes before serving.