In a medium bowl, whisk together olive oil, lime juice, 1/4 cup cilantro, minced garlic, cumin, 1/2 teaspoon salt, and black pepper.
Place chicken breasts in a shallow dish and pour the marinade over them; ensure even coating and let sit for at least 15 minutes.
Preheat a grill pan or outdoor grill to medium-high heat, approximately 400 degrees Fahrenheit.
Place chicken on the grill and cook for 6 to 8 minutes per side until the internal temperature reaches 165 degrees Fahrenheit.
While chicken cooks, combine diced avocados, tomato, red onion, the remaining 1/4 cup cilantro, and the remaining 1/2 teaspoon salt in a small bowl.
Remove chicken from the grill and allow it to rest for 5 minutes to retain juices.
Spoon the avocado salsa generously over each chicken breast and serve immediately.