Season the chicken breast with 2 tablespoons balsamic vinegar, Italian seasoning, salt, and black pepper. Cover and allow to rest on the counter as the grill heats, no longer than 1 hour.
Prepare the grill according to the manufacturer's directions to burn at medium-high heat.
Place the chicken on the grill and cook, uncovered, for 5 to 8 minutes on each side depending on the thickness. Test the chicken for an internal temperature of 165 degrees.
Remove the chicken and top each chicken breast with a slice of mozzarella. Tent the platter with aluminum foil and allow the cheese to melt.
Meanwhile, place the remaining balsamic vinegar into a small saucepan with the brown sugar. Bring the mixture to a boil over medium-high heat and immediately reduce the temperature to medium-low, allowing the vinegar to reduce by half. The vinegar will thicken as it cools. If the mixture becomes too thick to spoon over the chicken, add 2 teaspoons of water and return to the heat, stirring until the mixture is mixed and begins to thicken again.
To serve, remove the foil and top the cheese with a slice of tomato and sprinkle of fresh basil. Drizzle with the balsamic reduction.