Heat balsamic vinegar in a small saucepan over medium heat. Bring vinegar just to a boil then reduce heat slightly and allow to simmer, stirring occasionally, until vinegar has reduced by half, about 10 minutes. Remove from heat and set aside.
Brush grill grates lightly with oil and pre-heat grill to medium-high heat. In a small bowl, stir together olive oil and garlic.
Brush each side of each chicken breast lightly with olive oil mixture and season each side with salt and pepper.
Transfer chicken to preheated grill, cover with lid and cook chicken 5 minutes, then rotate and cook opposite side until chicken is cooked through and registers 165 degrees on an instant read thermometer (about 5 minutes longer, time will vary though based on the thickness of the chicken breasts).
Top each chicken breast with one slice of fresh mozzarella and 1 or 2 Roma tomato slices. Transfer to a platter and cover with foil, allow to rest 5 - 10 minutes.
Remove foil and sprinkle tops with fresh basil and drizzle with balsamic reduction, serve immediately.