Prepare a gas or charcoal grill to high heat.
Shape the dough into 2 medium pizzas while on a clean floured surface. Let the dough sit for 5 minutes and then re-form to make sure it's as big as you'd like. Place the pizza dough on a lightly floured rimless baking sheet, or pizza peel.
Brush one side of the dough with olive oil. Transfer the dough, olive oil side down, to the grill, and transfer the pizza dough directly onto the grill. Grill the dough for about 2 minutes with the lid closed. Drizzle the top side with olive oil. Using a pair of tongs flip the dough over onto the other side and cook for about 2 minutes more. The dough should be golden brown with grill marks on both sides.
Once the dough is cooked, remove it from the grill and place it back onto the baking sheet or pizza peel.
Brush more olive oil over the pizza and sprinkle with garlic. Arrange the mozzarella on top and place back onto the grill until the cheese has melted.
Using tongs or a peel, remove the pizza from the grill and top with sliced peaches, prosciutto, and basil. Drizzle with oil and season with salt and pepper as needed. Drizzle with reduced balsamic if desired. Slice and serve immediately.