In a small bowl, whisk together olive oil, balsamic vinegar, minced garlic, oregano, salt, and pepper.
Place chicken breasts in a shallow dish and coat thoroughly with the marinade; allow to rest for 15 minutes.
Preheat the grill to medium-high heat (approximately 400°F or 200°C).
Place chicken on the grill and cook for 6 to 7 minutes per side until the internal temperature reaches 165°F (74°C).
During the final 2 minutes of grilling, place one slice of mozzarella cheese on each chicken breast and close the grill lid until melted.
Remove chicken from the grill and top each breast with sliced avocado and tomato.
Garnish with fresh basil chiffonade and drizzle with balsamic glaze immediately before serving.