Cut the tomatoes into ½-inch dice (you should have 2 cups). Put them in a fine-mesh sieve, shake it over the sink to get rid of excess juice, then dump the tomatoes into a medium bowl.
Mince, press, or finely grate two of the garlic cloves and add them to the tomatoes. Add 2 tablespoons of olive oil, the balsamic vinegar of Modena (a.k.a. thinner balsamic), chopped basil, salt, and pepper. Mix with a fork and let marinate at room temperature for about 30 minutes.
Mince, press, or finely grate the remaining garlic clove and put it in a ramekin or small bowl. Add the remaining 2 tablespoons of oil, mix with a fork, and let the oil infuse at room temperature for about 30 minutes.
Preheat the oven to 500°F (if using a baking sheet) or as high as it will go (if using a baking stone/steel; place the stone in the middle or place the steel in the bottom third of the oven before you start preheating). Let the oven preheat for at least 30 minutes. Then, if you’re using a baking stone or steel, switch the oven to Broil on high.
Stretch or roll out your dough to a 12- to 14-inch circle, then transfer it to a baking sheet / pizza pan or a lightly floured pizza peel (if using a baking stone/steel).
Brush the dough with the garlic oil (you won't need it all; just make sure you cover the whole surface, all the way to the edges). Use a slotted spoon or your hands to add the tomato mixture, leaving the accumulated liquid in the bowl. Spread the tomatoes evenly, then top with the diced fresh mozzarella and finely shredded parm. Season with a pinch or two of salt and a few grinds of black pepper.
Transfer the pizza to the oven and bake until the crust is evenly browned on the bottom and the cheese has browned in spots—8 or 9 minutes on the baking sheet, 5 to 8 minutes on the baking stone/steel.
Take the pizza out of the oven. Drizzle it with the thicker aged balsamic vinegar (or balsamic glaze), then tear the fresh basil leaves and scatter them on top. Slice and serve.