Add the chicken breasts to a large zip to lock bag. Use a rolling pin or meat mallet to pound the breasts until they are fairly even in diameter. Add marinade, seal and move contents around until the chicken is evenly coated. Allow to marinate at room temperature up to one hour or place in the refrigerator to marinate longer.
Heat 2 tablespoons olive oil in a large cast iron skillet over medium heat. Place chicken breasts in hot skillet and cook for 7-8 minutes. Flip over and cook until chicken is cooked completely, another 6-8 minutes, or until chicken is cooked through.
Remove chicken from pan and cover with foil to keep warm and moist..
Increase heat on skillet to medium-high and add remaining olive oil to skillet. Add the sliced peppers and onion and cook, stirring frequently, until crisp tender and nicely browned, approximately 6-7 minutes.
Cut the chicken into thin slices and serve with peppers, onions, sliced avocado, lime wedges, and fresh cilantro on the side.