Mince garlic in a food processor. Add one-third basil and pulse until fine. Blend in 1/2 cup olive oil until light yellow. Add 1/2 cup balsamic vinegar and pulse to combine.
Place chicken in a resealable bag with salt, pepper, and basil marinade. Seal and refrigerate for 2 hours or overnight.
Preheat oven to 300°F (150°C).
Brown chicken in a hot skillet for 4–5 minutes per side. Transfer to a baking dish.
Bake for 15–20 minutes until fully cooked. Slice into strips and return to dish to absorb juices.
Cook penne in salted boiling water until al dente. Drain.
In a large bowl, combine sliced basil, tomatoes, remaining olive oil, and balsamic vinegar. Season to taste.
Add pasta and chicken with juices. Toss gently to combine and serve warm or chilled.