Preheat oven to 375F and spray baking dish with cooking spray.
Combine 1/4 cup balsamic vinegar and olive oil in a bowl. Marinade the chicken in balsamic mixture for 15 minutes.
Place chicken breasts in baking dish, pour half of the mixture over the meat, and season with salt and pepper.
Bake at 375F until chicken is cooked through (about 30 minutes), occasionally spooning the balsamic mixture back over the chicken.
During the last 2 minutes of baking, place 2 slices of cheese over the top of each chicken breast to melt.
In a large bowl combine the tomatoes, basil, garlic, a pinch of salt and remaining balsamic. Toss gently to combine.
Place the chicken breasts on plates, top with tomato mixture and serve.