Place chicken in a shallow dish or large resealable plastic bag. Pour the dressing over it, cover or seal, and marinate in the refrigerator for 12 to 24 hours.
Coat a large skillet with cooking spray or oil, and set over low heat. Remove the chicken from the marinade and discard the marinade. Fry chicken breast halves over low heat for about 30 minutes, or until juices run clear.
Arrange chicken on a serving platter. Place a generous slice of fresh mozzarella on top of each piece. Place a leaf of basil on top of the cheese, and cover with a slice of tomato. Dash balsamic vinegar over the platter, and season with salt and pepper. Serve!