In a medium bowl, whisk together 3/4 cup balsamic vinegar, 1/4 cup honey, 3 minced garlic cloves, 2 tablespoons olive oil, 2 teaspoons Italian seasoning, 1/2 teaspoon salt, and 1/4 teaspoon black pepper to make the marinade.
Add the 4 chicken breasts to the marinade, coat well, and refrigerate to marinate for 30 minutes.
While the chicken marinates, combine 1 sliced avocado, 1/2 cup sliced cherry tomatoes, and 1/2 cup basil pesto in a small bowl; season with salt and pepper to taste and set aside.
Heat a grill to medium-high or preheat a large skillet over medium-high heat for stovetop cooking.
If grilling: cook the chicken 6 minutes per side or until no longer pink and internal temperature reaches 165°F; if using a skillet: discard excess marinade but reserve 1/4 cup, then cook chicken 3–4 minutes per side until cooked through.
If using the skillet, add the reserved 1/4 cup marinade to the pan and simmer until the sauce reduces slightly.
Top each cooked chicken breast with a slice of mozzarella, spoon the avocado-pesto-tomato mixture over the cheese, and finish with 1/4 cup chopped fresh basil (divided across servings).