Season chicken breasts with kosher salt and fresh cracked black pepper. Add a couple of tablespoons of extra virgin olive oil to your skillet. This may seem like quite a bit, but it will form the base for the sauce.
While waiting for the oil to heat up, go ahead and get your water going for the pasta. When the oil is HOT, add the chicken breasts. After about 4-5 minutes, the chicken breasts should be ready to turn. By the way, when did chicken breasts get SO HUGE?! You'll want to cook them on the second side another 4-5 minutes, or until the juices run clear when you pierce them with a small sharp knife. You don't want to OVER cook them, or they will be dry and tough. If you pounded yours out thinner, they will cook faster than this.
After the second side has been cooking for about 4 minutes, go ahead and add in the fresh tomatoes that you have chopped up. Next, add the fresh basil that you have chopped into ribbons, the cold butter, the minced garlic, and you will probably want to add some more salt and black pepper to season the sauce. Now give it all a good stir so all those flavors will marry together.
When your chicken is "done," turn the sauce down on the lowest setting to keep it all warm until the pasta is cooked, which should be any minute now. At this point, if still waiting for the pasta, you can remove the cooked chicken to a cutting board, and after resting just a couple of minutes, cut the chicken into slices on the bias.
Divide the pasta among individual bowls, and top each with half of a chicken breast, sliced into pieces, and equal portions of the tomato basil sauce. Top with more fresh basil and freshly grated parmesan cheese.