Start by placing your minced garlic cloves in a food processor or blender. Pulse until they're finely chopped, then add about one-third of your fresh basil leaves. Continue pulsing until the basil is finely minced and fragrant. Slowly drizzle in half a cup of olive oil while processing, watching as the mixture transforms into a beautiful light green paste with flecks of basil throughout. Finally, add half a cup of balsamic vinegar and process just until everything is combined into a smooth, aromatic marinade.
Place your chicken breasts in a large resealable plastic bag along with the tablespoon of salt and 1 1/2 teaspoons of black pepper. Pour the basil marinade over the chicken, making sure to squeeze out as much air as possible before sealing the bag. Massage the marinade into the chicken through the bag, ensuring every piece is well-coated. Refrigerate for at least 2 hours, though overnight marinating will give you even deeper flavors. I like to flip the bag halfway through marinating to ensure even coverage.
Preheat your oven to 300°F while you heat a large skillet over high heat. Remove the chicken from the marinade (save any leftover marinade) and brown each piece for 4-5 minutes per side until you get a beautiful golden crust. The chicken will look quite dark from the balsamic vinegar, but don't worry - this creates incredible flavor. Transfer the browned chicken to a large baking dish and bake for 15-20 minutes until the internal temperature reaches 165°F.
While the chicken is baking, bring a large pot of salted water to a rolling boil. Add your penne pasta and cook according to package directions until al dente, usually about 11 minutes. During the last few minutes of cooking, slice your remaining basil leaves into thin ribbons. In a large serving bowl, combine the halved grape tomatoes with the sliced basil, remaining 1/4 cup olive oil, and remaining 1/4 cup balsamic vinegar. Season this mixture with salt and pepper to taste, then let it sit to allow the flavors to meld.
Once the chicken is cooked through, transfer it to a cutting board and slice into 1/4-inch strips. Return the sliced chicken to the baking dish to absorb any remaining cooking juices. Drain the pasta well and immediately add it to the tomato and basil mixture. Top with the sliced chicken and all the delicious pan juices, then toss everything together until well combined. The heat from the pasta will slightly warm the tomatoes and help all the flavors come together beautifully.