Set a pizza stone or heavy baking sheet in the bottom third of the oven. Preheat the oven to 550°F, allowing at least 30 minutes for the stone to heat.

On a lightly floured work surface, roll 1 ball of dough into a 6-inch round. Turn the edges of the dough up slightly to form a lip, then stretch the dough out to a 10-inch round. Transfer the dough to a lightly floured pizza peel or upside-down baking sheet.

In a food processor, pulse the tomatoes with the basil, garlic, and olive oil until finely chopped. Season with salt and pepper.

Spread half of the tomato sauce on the pizza dough to within 1/2 inch of the rim. Slide the pizza onto the hot stone and bake for 5 minutes, or until the sauce is slightly dry and the crust is lightly golden. Remove the pizza from the oven, scatter half of the mozzarella over the top, and return to the oven. Bake for 4 minutes longer, or until the cheese is bubbling and the crust is golden and crisp. Let the pizza rest for 5 minutes before cutting into wedges or squares. Repeat with the remaining dough, sauce, and cheese to make the second pizza.