Preheat oven to 425 degrees.
Cut chicken breasts in half horizontally so that each piece of chicken is about 1/2 inch thick. Note: If you only have one large cast iron skillet, you will likely only be able to fit 6 chicken pieces (3 breasts). If you have another skillet or want to do two batches, this recipe will work for 8 chicken pieces (4 breasts)
In a small bowl, stir together salt, pepper, dried oregano, and thyme. Evenly coat both sides of each piece of chicken.
Heat a large cast iron skillet over medium-high heat. Add chicken and sear on each side for about 1 minute, or until lightly browned. Remove chicken from skillet after it has been seared.
Turn heat to low and melt butter. Add garlic and saute for 1 minute. Stir in balsamic vinegar and brown sugar. Once will combined, add chicken. Spoon sauce over chicken so that both sides of chicken are coated.
Spread tomatoes evenly around chicken. Cover skillet with aluminum foil. Bake for 15 minutes or until chicken reaches 165 degrees.
Remove from oven and cover each piece of chicken with a slice of mozzarella cheese. Turn oven to broil. Return skillet to oven and broil until cheese is melted and tomatoes have blistered, about 3 minutes. NOTE: Tomatoes will likely spread out juice as they pop. Place skillet on a large baking sheet or baking mat in oven. As you broil, keep oven minimally open to prevent getting sprayed with hot tomato.
Remove skillet from oven. Serve chicken with juices from pan and tomatoes. Top with fresh basil and balsamic glaze.