In a large bowl, combine one-third of the oil and half of the lime juice with the cumin, chili powder, garlic powder, onion powder, two-thirds of the salt and pepper. Whisk to thoroughly mix. Add the chicken breasts to the marinade and toss to evenly coat. Cover, place in the refrigerator and let the chicken marinate for at least 1 hour and up to four hours.
Meanwhile, make the salsa. In a medium bowl, combine the avocado, tomato, feta, capers, shallot, cilantro, one-third of the oil and remaining lime juice. Season with remaining salt and pepper and lightly toss to combine. Refrigerate salsa for at least one hour to allow the flavors to come together.
When ready to cook, heat the remaining oil in a large sauté pan over medium heat until oil shimmers. Add chicken, smooth skin-side down. Cook, without moving, until the skin is deep golden brown and chicken easily releases from the pan, 8 to 10 minutes. Turn chicken and cook until a thermometer inserted into the center of breast registers 160°F, 6 to 8 minutes. Remove and let the cooked chicken breasts rest a few minutes before cutting into thin slices and seasoning with a bit of salt and pepper.
Served sliced chicken topped with avocado salsa.