Place the the flat rack over the roasting pan. Lightly grease the top of the rack with the butter or oil.
Place the chicken leg quarters on the top of the flat rack.
Bake for approximately 3 ½ to 5 hours, or until the chicken reaches an internal temperature of 165 degrees. The longer you cook the legs to the 5-hour mark, the more they are going to be falling off of the bone. (You MUST check the water level in the bottom of the roasting pan at least every 30 minutes throughout the entire process. This recipe method is meant to smoke very, very gently. Just a few wisps of smoke escape that inject the chicken quarters with the wood flavor. If the pan is smoking too much, a fire could break out. So, please check the water level in the bottom of the pan at least every 30 minutes. Add enough water to fully saturate the wood chips, and keep a very small stream of smoke rising out of the bottom of the pan.)
(If not serving immediately, you can remove the chicken at 160 degrees and let it rest up to 165 degrees before serving.)
During the last hour of oven-smoking, baste the chicken with sauce. Let cook on one side with sauce for 30 minutes, then turn each piece over, baste, and cook for an additional 30 minutes.