In a large bowl, combine the chicken breasts with balsamic vinaigrette, oregano, garlic powder, salt, and pepper. Marinate for at least 10 minutes.
Preheat an outdoor grill or indoor grill pan to medium-high heat (approximately 400°F).
Place chicken on the grill and cook for 5 to 6 minutes per side, or until the internal temperature reaches 165°F.
During the final 60 seconds of grilling, place one slice of mozzarella cheese on each chicken breast and cover the grill to melt.
Remove the chicken from the heat and let rest for 3 to 5 minutes to retain juices.
Top each chicken breast with several slices of fresh California avocado.
Drizzle with balsamic glaze and garnish with fresh basil before serving.