In a bowl, combine all the rub ingredients and thoroughly coat the chicken breasts with this mixture. If you have the time, let the chicken marinate for at least 30 minutes at room temperature or refrigerate up to 8 hours then bring it back to room temperature before cooking. While the chicken marinates, start preparing the avocado salsa.
For stovetop cooking, if the chicken has been chilled, leave it out for 15 to 30 minutes to reach room temperature. Heat a grill pan or skillet with a bit of grease over medium-high heat until hot. Place the chicken in the pan and cook without turning for 3 to 5 minutes, aiming for a nice browned exterior.
Flip the chicken, cover the pan, lower the heat to medium, and cook for another 5 to 7 minutes, depending on thickness, until cooked through. Remove the chicken and allow it to rest for 5 minutes before slicing. Add extra fresh lime juice to garnish if you like.
For grilling, preheat your grill to medium heat (375-450°F) and grease the grates. After bringing the chicken to room temperature, grill it without moving for 5 to 7 minutes per side until the internal temperature reaches 165°F. Let the chicken rest 5 minutes before slicing.
To make the avocado salsa, mix all ingredients except the avocados in a large bowl, cover, and chill from 30 minutes up to 24 hours to blend flavors. Just before serving, carefully fold in the chopped avocados and adjust seasoning by adding salt, pepper, or jalapeno seeds for extra heat if desired.