Add oil to cover the cast iron skillet and bring to medium heat. Season the chicken breasts with salt and pepper. Cook chicken in the skillet, covered, for about 6-8 minutes on each side. Remove the chicken and place on a plate.

Add balsamic and bring the heat to medium-low heat. Add the cherry tomatoes and toss them with a slotted spoon. Cook for about 5 minutes. Halfway through the 5 minutes add basil, garlic, and salt and mix with the spoon.

Add back in the chicken breasts and place the sliced mozzarella on top. Cover the skillet to assure the cheese melts. Top each breast with the balsamic "sauce" with tomatoes, and serve hot!