Stir 1 teaspoon salt, 1/4 teaspoon black pepper, and cayenne pepper together in a small bowl; rub evenly into the chicken breasts.
Heat the olive oil in a skillet over medium heat; cook the chicken in the heated oil until no longer pink in the center and the juices run clear, 5 to 7 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Stir the onion, lime juice, and avocado together in a bowl; season with salt and pepper. Spoon over the chicken breasts to serve.