
Preheat grill to medium-high or heat a grill pan over medium-high heat.
Combine oil, 1 tablespoon balsamic glaze, basil, garlic, pepper and salt in a small bowl. Brush the mixture on both sides of chicken.
Oil the grill rack or pan. Grill the chicken until it easily releases from the grill or pan, 3 to 4 minutes. Flip and top with mozzarella and tomato. Grill until the cheese is melted and an instant-read thermometer inserted into the thickest part of the chicken registers 165°F, 3 to 4 minutes more. Drizzle with the remaining 1 tablespoon balsamic glaze and sprinkle with more basil, if desired.