While the olive oil heats, pat the chicken breasts dry and lightly salt and pepper both sides.
Fry the chicken over medium high heat, turning occasionally until it is evenly golden on both sides, about five minutes It will not be cooked through yet.
Pour the balsamic vinegar over the cooking chicken. You don't want to boil away the vinegar, so you may need to lower the heat to keep it from bubbling.
Add the tomatoes and cover, cooking over medium heat until the tomatoes begin to split, about 5-6 minutes. You want the tomatoes to be soft but do not want to crush them into the pan.
When the tomatoes are ready, top each piece of chicken with one slice of mozzarella and cover, letting the cheese melt at medium heat.
Test your chicken to be sure it is cooked. If you have an instant read thermometer, it will read 165 degrees F.