Grill or roast chicken cutlets. Cut into strips and set aside.
In the meantime, bring water to boil in a large stockpot that has been seasoned with about 3 tablespoons of olive oil and and about 1/2 teaspoon of salt. Once it has reached a rolling boil, add pasta noodles and stir. Cook for about 9-10 minutes. Remove from heat and strain noodles through a colander. Rinse with cold water to cool.
In a large bowl, add onion, tomatoes, pasta, and chicken.
In a small sauce pan, combine the balsamic vinegar, oil and salt and pepper. Cook over low to medium heat until it reduces by half and is much thicker. Pour over pasta and toss well.
Add cheese and fresh basil to pasta. Toss and serve.