In a medium bowl, whisk together the marinate ingredients. Place chicken into a large resealable bag and pour in the marinate. Toss to coat and store in the fridge for 8 hours (4-8 is fine, but flavor will not be as strong). Heat grill to medium heat. Remove chicken from marinate and grill chicken 3-5 minutes per side or until no longer pink (this will depend on size and thickness of the chicken breast). During the last minute of cooking place two slices of cheese on the chicken and allow it to melt. Remove chicken from grill one the cheese has melted. To serve, plate the chicken and spoon the tomato mixture over the chicken. Top with a bit of the reserved basil and serve immediately.
In a medium pan, slowly simmer ⅔ cup balsamic vinegar and granulated sugar till reduced by a third. Add tomatoes and basil and cook over low heat while you prepare the chicken. It is important to keep the heat low enough once the vinegar has reduced to the tomatoes and basil cook slowly.
Heat grill to medium heat. Remove chicken from marinate and grill chicken on both sides until the internal temperature reaches 165 degrees F. During the last minute of cooking place two slices of cheese on the chicken and allow it to melt. Remove chicken from grill one the cheese has melted.
Top chicken with tomato mixture and reserved chopped fresh basil. Enjoy.