In a medium bowl, whisk together 3/4 cup balsamic vinegar, 2 tablespoons honey, 2 tablespoons extra virgin olive oil, 2 teaspoons Italian seasoning, 1 teaspoon garlic powder, 1/2 teaspoon salt, and 1/4 teaspoon ground black pepper to make the marinade.
Place 4 boneless skinless chicken breasts in a large resealable bag, pour in the marinade, seal, and massage the bag so the chicken is evenly coated. Refrigerate for at least 30 minutes or up to 24 hours.
Just before grilling, combine 2 diced avocados and 1 pint quartered cherry or grape tomatoes in a bowl; season with a pinch of salt and pepper and toss gently. Keep chilled until ready to top the chicken.
Preheat a grill or grill pan to medium-high and oil the grates if needed.
Grill the chicken breasts 6–8 minutes per side, until cooked through (no pink in the center and juices run clear).
Top each cooked breast with a slice of mozzarella, cover the grill or close a lid, and cook 1–2 minutes more until the cheese melts.
Transfer the chicken to a platter, spoon one quarter of the avocado-tomato mixture onto each breast, sprinkle with 2 tablespoons sliced basil, and finish with a drizzle of balsamic glaze before serving.