In a bowl, whisk together the balsamic vinegar, olive oil, garlic powder, oregano, salt, and pepper. Place the chicken breasts in a resealable plastic bag or shallow dish and pour the marinade over them. Seal the bag or cover the dish, and refrigerate for at least 30 minutes, or up to 2 hours for more flavor.
Preheat your grill to medium-high heat. Remove the chicken from the marinade and discard the marinade. Grill the chicken for 6-7 minutes on each side, or until the internal temperature reaches 165°F. Remove from the grill and let it rest for 5 minutes.
While the chicken is resting, prepare the Caprese salad. In a mixing bowl, combine the halved cherry tomatoes, diced mozzarella, and chopped basil. Drizzle with olive oil and season with salt and pepper to taste. Toss gently to combine.
Slice the grilled chicken and serve it topped with the Caprese salad. Drizzle with balsamic glaze if desired.