Add chicken and marinade ingredients to a large zip bag or bowl. Let chicken marinate for 15-20 minutes.
Make the avocado salsa:
While the chicken marinates make the avocado salsa in a medium bowl by combining the avocados, shallot, cilantro, juice of 1 ½ limes, and salt in a bowl. Mix with a fork and refrigerate until ready to use.
Cook the chicken breasts:
Heat a large pan or cast-iron skillet over medium-high heat. Once hot, heat the oil and distribute it around to coat. Place chicken breasts in the pan, discard the marinade, and cook for about 6 minutes on the first side. Do not move the chicken around. Flip and cook the other side for an additional 4 to 6 minutes until the internal temperature reaches 165F. Remove from the pan onto a plate.
Preheat the grill to medium-high heat (375-400F). Once hot, grease the grates with oil. Place the marinated chicken breasts on the hot grill, and cook with the lid closed for 6-8 minutes. Open the lid, flip the chicken using tongs, and cook for 6 more minutes, until the internal temperature reaches 165F. Remove from the grill.
Serve:
Serve the chicken breasts topped with avocado salsa. Sprinkle with additional chopped cilantro and squeeze more lime over the top (if desired).