Heat oven to 400°F. Heat oil in large ovenproof skillet on medium-high. Season chicken with ½ teaspoon salt and ¼ teaspoon pepper and cook until deep golden brown on 1 side, 4 to 5 minutes.
Flip chicken and cook 1 minute. Reduce heat to medium, add balsamic vinegar and gently simmer until slightly thickened and syrupy, 1 to 2 minutes. Transfer skillet to oven and roast 10 minutes.
Turn chicken to coat in vinegar, top with mozzarella and roast until chicken registers 165°F on instant- read thermometer and cheese begins to melt, about 2 minutes.
Serve topped with basil leaves and tomato and sprinkle with salt and pepper. Drizzle with any additional balsamic glaze from skillet if desired.
