Place the chicken breasts in a large zip-top bag or shallow bowl.
In a small bowl, whisk together the lime juice, olive oil, cilantro, cumin, garlic, and salt.
Pour the marinade over the chicken and coat evenly. Refrigerate for 15–20 minutes.
While the chicken marinates, prepare the avocado salsa. Combine the diced avocados, shallot, cilantro, lime juice, and salt in a medium bowl. Gently mix and refrigerate until ready to serve.
To cook on the stovetop, heat a large skillet or cast-iron pan over medium-high heat. Add olive oil and swirl to coat.
Remove the chicken from the marinade and discard the remaining marinade.
Place the chicken in the hot skillet and cook for approximately 6 minutes on the first side without moving it.
Flip and cook for an additional 4–6 minutes or until the internal temperature reaches 165°F (74°C).
For grilling, preheat the grill to medium-high heat (375–400°F). Lightly oil the grates.
Place the marinated chicken on the grill and cook with the lid closed for 6–8 minutes.
Flip using tongs and continue grilling for another 6 minutes or until fully cooked.
Let the Cilantro Lime Chicken Breasts rest for 5 minutes before serving.
Top each chicken breast with a generous spoonful of avocado salsa and serve immediately.