Prepare a gas or charcoal grill for indirect heat; clean and lightly oil grates. Season the tomatoes with ½ teaspoon salt. Working with one piece of dough at a time, gently stretch or roll each piece to a 10 to 12-inch round (it doesn’t need to be perfect, so a little oval or square is OK). Place round on a baking sheet, brush both sides with olive oil, and season with salt and pepper.
Quickly and confidently place dough on hot grates over direct heat.
Grill, undisturbed, until bubbles appear on the surface of the dough, about 2 minutes.
Using tongs and a wide metal spatula, flip dough and place over indirect heat. Top with crushed tomatoes.
Add mozzarella and season with salt and pepper.
Cover and grill until cheese is melted and underside of dough is lightly charred, 4 to 5 minutes more (the time it takes will vary as your coals die, so keep an eye on it).
Using tongs and a spatula, transfer pizza to a board and top with basil leaves, crushed red pepper flakes and a drizzle of olive oil.