Combine the halved or quartered cherry tomatoes, 4 minced garlic cloves, 3 tablespoons chopped basil, diced shallot (if using), 2 teaspoons olive oil, and a pinch of salt and pepper in a bowl; stir and let sit at room temperature while you prepare the patties.
Preheat your grill or grill pan to high heat.
In a mixing bowl, place 1 pound ground chicken or turkey and add 2 tablespoons chopped fresh basil, 1 teaspoon dried basil, 1 teaspoon garlic powder, 1/2 teaspoon salt, 1/2 teaspoon freshly ground pepper, and 1/4 teaspoon crushed red pepper; mix gently until combined without overworking.
Divide the mixture into 6 portions (or 4–6 to match your buns) and form each portion into a patty pocket: press a 1/2-inch mozzarella cube into the center and wrap the meat around it so the cheese is fully enclosed.
Place the stuffed patties on the preheated grill and cook about 5 minutes on the first side, then flip and cook 3–5 minutes more until an instant-read thermometer inserted into the meat (not the cheese) reads 165°F.
When the burgers are nearly done, place additional mozzarella on top of each patty and close the grill or cover briefly so it melts, about 1 minute.
Assemble burgers by placing each cooked patty on a bun, spooning the tomato-basil mixture over the patty, and drizzling with balsamic glaze; serve immediately.