Begin by making the Salsa Verde: Process all the ingredients in the food processor until a thick mixture is formed. Set aside.
Next, in a large skillet over medium heat, cook the peppers and onions in a tablespoon of the oil. Cook them until they're crisp tender, about ten minutes. Remove them from the skillet and set aside. Cover to keep warm.
Meanwhile, cut each breast horizontally into three pieces. Pound each piece until they are evenly thin.
Mix all the spices together and coat both sides of the chicken in the spices.
Reheat your skillet to medium heat, and add the rest of the oil. Cook the chicken on both sides, until it's cooked through. Cut the cooked chicken into half inch pieces and set aside. Cover and keep warm.
Heat a cast iron skillet on high heat. Toast each tortilla on both sides.
Fill your fajitas with chicken, onions, peppers, salsa verde, sour cream, and avocado.