Whisk brown sugar, soy sauce, teriyaki sauce, and barbeque sauce together in a large glass bowl until sugar is dissolved. Stir black food coloring in small amounts into the mixture until you get the desired dark color. Liberally pierce the wings with a fork; add to the sauce.
Cover the bowl with plastic wrap and marinate in the refrigerator at least 1 hour to overnight.
Preheat oven to 350 degrees F (175 degrees C).
Pour the chicken and sauce into a large baking dish.
Bake the chicken in the preheated oven until no longer pink at the bone and the juices run clear, about 1 hour. An instant-read thermometer inserted into the thickest part of the thigh, near the bone should read 165 degrees F (74 degrees C).
