large beefsteak tomatoes (about 3 lb. total), 3 grated on the large holes of a box grater, 1 coarsely chopped
cloves garlic, finely grated
kosher salt, plus more
granulated sugar
extra-virgin olive oil, divided
fusilli
boneless, skinless chicken breasts (about 1 1/2 lb. total)
Freshly ground black pepper
balsamic vinegar
brown sugar
mozzarella, torn into bite-size pieces
fresh basil leaves, torn