First, fill a large pot with water and bring it to a boil over high heat. Add penne pasta and cook until the pasta is al dente. Strain the pasta and set it aside until later.
Heat olive oil in a large Dutch oven over medium/high heat.
When olive oil is fragrant, add the chicken, garlic, 1/4 teaspoon salt, and 1/8 teaspoon pepper to the pot and cook for 4-5 minutes.
Next, when the chicken is partially cooked, add cherry tomatoes to the pot and cook the tomatoes, chicken, and garlic together until the tomatoes begin to stew (about 5 minutes).
Add the balsamic vinegar, white wine, butter, and the rest of the salt and pepper to the pot, mix well, and let the liquid reduce by half (about 10 minutes).
Remove the sauce from the heat and toss the noodles and mozzarella balls with the sauce.
Add basil to the pasta right before serving, toss, and enjoy.