In a small bowl, combine 1 teaspoon salt, 1/4 teaspoon pepper, and cayenne; rub all over chicken.
In a large skillet, heat oil over medium. Add chicken, and cook until browned on the outside and opaque throughout, 8 to 10 minutes per side.
Meanwhile, in a medium bowl, combine onion and lime juice; set aside. Just before serving, fold avocado chunks into onion mixture; season with salt and pepper. Serve chicken topped with salsa.