1. Preheat the broiler. Place the chicken pieces, skin-side up, in a large cast-iron skillet or broiler tray and sprinkle on the soy sauce, balsamic vinegar, garlic, and hot-pepper sauce, if using.
2. Broil the chicken pieces for 10 minutes, about 6 inches from the flame. Flip and broil for another 10 minutes. Turn pieces again and broil for 5 to 10 minutes more, or until the skin is crisp and the meat is cooked through.
3. Remove the chicken to a warm platter and cover with foil. Place the skillet or tray on the stove over moderate heat. Skim off most of the excess fat with a spoon (a little fat adds flavor). Stir the flour into the remaining juices to create a thick paste; cook for 1 minute.
4. Slowly pour in the broth or water, whisking to create a smooth gravy. Let the gravy simmer for 3 to 5 minutes, or until the chicken flavors have intensified and the gravy is thick enough to coat a spoon. Season to taste. Serve by pouring over the hot chicken or into a gravy boat.
How kids can help: Sprinkle the chicken with the soy sauce, balsamic vinegar, garlic, and hot-pepper sauce.