Stir together 1 teaspoon salt, 1/4 teaspoon black pepper, and cayenne pepper in a small bowl until well combined; rub evenly into chicken breasts.

Heat olive oil in a large skillet over medium heat; cook chicken in hot 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 together onion, avocado, and lime juice in a medium bowl until combined; season with salt and pepper. Spoon over chicken breasts to serve.