Place chicken in a large resealable bag. In a large bowl, whisk together the next 5 ingredients (coconut aminos through salt and pepper) and pour the mixture into the bag. Seal bag and place chicken in the fridge for at least 2 hours.
Heat grill to medium heat.
Thread chicken onto large skewers. Place skewers on the grill and cook for 5 to 10 minutes on each side until chicken is no longer pink inside.
In a large bowl, toss the remaining 6 ingredients (broccoli through lime juice) with sea salt and freshly ground black pepper to taste. Serve chicken with salad.