In a small bowl, whisk together the balsamic vinegar, olive oil, minced garlic, Italian seasoning, salt, and pepper.
Place the chicken breasts in a large resealable bag or a shallow dish. Pour the marinade over the chicken, making sure each piece is well coated. Seal the bag or cover the dish and refrigerate for at least 30 minutes (or up to 4 hours for more flavor).
Preheat your oven to 400 degrees F.
Transfer the marinated chicken breasts to a baking dish. Pour any leftover marinade over the chicken.
Scatter the halved cherry tomatoes and onions around the chicken in the baking dish.
Bake in the preheated oven for about 25-30 minutes, or until the chicken is cooked through and the internal temperature reaches 165°F (75°C).
Remove the baking dish from the oven and sprinkle the shredded mozzarella cheese evenly over the chicken breasts.
Return the dish to the oven and bake for an additional 5-7 minutes, or until the cheese is melted and bubbly.
Sprinkle fresh basil leaves over the top before serving for a burst of fresh flavor.
Serve the balsamic baked chicken with your favorite sides like roasted vegetables, a simple salad, or garlic bread.