In a medium bowl, whisk together balsamic vinegar, honey, minced garlic, olive oil, Italian seasoning, salt, and pepper until well combined.
Pour marinade into a gallon-sized ziplock bag. Add chicken breasts, seal bag, and massage to coat evenly. Refrigerate for at least 30 minutes or up to 1 hour.
While chicken marinates, dice avocados and tomatoes. Toss together in a small bowl with a pinch of salt and pepper. Set aside.
Preheat grill to medium-high heat (about 400°F). Lightly oil grates with a paper towel dipped in oil.
Remove chicken from marinade, letting excess drip off. Place on hot grill and cook for 6 minutes per side, or until internal temperature reaches 165°F.
Immediately place a slice of mozzarella cheese on each chicken breast. Transfer to a platter and let rest for 5 minutes.
Top each chicken breast with avocado-tomato mixture. Sprinkle with fresh basil and drizzle with balsamic glaze if desired. Serve immediately.