In a large bowl, thoroughly mix soy sauce, water, brown sugar, green onions, onion, garlic, sesame oil, and coconut milk. Add chicken pieces, pushing them down into the marinade to be sure all chicken pieces are covered. Loosely cover bowl with plastic wrap, and place in refrigerator to marinate 4 to 8 hours.
Toward the end of marinating time, preheat grill (or broiler).
Remove chicken from bowl, discard marinade, and place chicken on grill or broil pan. Grill (or broil) chicken about 12-15 minutes or until chicken is cooked through, turning about half way through cooking.
Garnish servings with reserved sliced green onions.
This is delicious served with coconut rice. To make coconut rice, replace half of the water in your rice recipe with coconut milk.