In a medium glass bowl, combine the chicken, balsamic vinegar, olive oil, garlic, oregano, salt and pepper. Stir to combine.
Using 2 pieces of foil per packet (each 12 inches long), form an X by overlapping the foil. Alternatively, use one sheet of heavy duty foil.
Divide the chicken mixture in half and place in the middle of the foil. Lay the whole basil leaves on top, and sprinkle the tomatoes over top of the chicken. Wrap the piece of foil over the chicken and tomato mixture, pinching the foil to seal. Repeat with the rest of the foil, chicken and tomatoes.
Place the foil package directly on the grill and cook until the chicken is just cooked through, about 10 minutes per side.
Alternatively, place the two foil packets on a baking sheets and cook in a 400 degree oven, about 10 minutes per side.
Carefully unwrap the foil (the steam will be hot!), remove the whole basil leaves with tongs and layer the mozzarella cheese on top of the chicken. Gently wrap again on place on the grill until the cheese is melted about 5 minutes.
Unwrap and sprinkle with basil. Season to taste with salt and pepper. Serve.