Place chicken on a cutting board, cover with a sheet of plastic wrap and pound thicker parts of chicken to an even thickness. Transfer chicken to a gallon size resealable plastic bag.
In a small mixing bowl, whisk together lime zest, lime juice, olive oil, honey, cilantro, 3 cloves minced garlic and season with 1 tsp salt and 1/2 tsp pepper. Pour marinade mixture over chicken in bag.
Seal bag while pressing air out, then rub marinade over chicken. Allow to rest in refrigerator 2 - 4 hours (rotating bag to opposite side and moving chicken around halfway through if possible to ensure it's evenly marinating).
Preheat a grill over medium-high heat (to about 425 - 450 degrees F). Brush grill grates lightly with oil then grill chicken until center registers 160 - 165 degrees on an instant read thermometer, about 4 minutes per side. Remove and let rest 5 minutes.
For the salsa:
In a medium mixing bowl, toss together avocado, tomato, onion, garlic and lime and season with salt and pepper to taste.
Spoon a generous amount over chicken and sprinkle with chopped cilantro. Serve warm.
Notes
To cook in a skillet: Pan sear the chicken over medium heat with a little olive oil, until it registers 165 degrees F in center (don't forget to turn halfway).
To cook in the oven: Bake the chicken at 400 degrees F in a greased 13 by 9-inch pan. About 20 minutes, or again to 165 degrees F.