Make the pesto: in a food processor or blender, pulse the ingredients for pesto: basil, parmesan cheese, pecans, garlic, salt and pepper. While motor is running, add in the oil. Continue to pulse until smooth. Set aside.
Whisk together the marinade ingredients for the chicken: balsamic vinegar, coconut aminos, and garlic. Marinate the chicken in a large container for at least 20 minutes, up until 4 hours.
Heat the grill to medium high heat (or prepare a charcoal grill).
Place chicken on the prepared grill and cook for 4-6 minutes (depending on thickness of the chicken breast). Flip over and repeat on the other side. Continuing to grill until chicken is no longer pink and internal temperature is 165. Remove and cover with foil until ready to eat.
Serve by spooning 2 tablespoons of the pecan pesto on top of each chicken breast over a bed of greens.