Place chicken in a shallow dish or large resealable plastic bag. Pour the vinaigrette over it, cover or seal and marinate in the fridge for 12 to 24 hours.
Coat a large frying pan lightly with oil, and set over low heat. Remove the chicken from the marinade and discard the marinade. Saute chicken breast fillets 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. Drizzle balsamic vinegar over the platter, then season with salt and pepper. Serve!