Heat grill to high (450° to 550°) with a charcoal area left clear or a gas burner turned off to make an indirect heat area. Make rub for chicken: Pulse basil, garlic, oil, salt, and pepper in a food processor until roughly chopped. Rub mixture all over chicken.
Make salsa: Stir together tomatoes, mozzarella, basil, oil, vinegar, garlic, salt, and pepper in a medium bowl to combine; let stand at least 30 minutes.
Grill chicken skin side down over direct heat until browned, 5 to 7 minutes. Move pieces to indirect heat area, turning over, and grill until cooked through, 25 to 30 minutes. Transfer chicken to a platter, tent with foil, and let rest 5 minutes. Spoon tomatoes, cheese, basil, and half the salsa liquid over chicken; save remaining liquid for a salad dressing if you like.
Note: Nutritional analysis is per serving.