Arrange the chicken quarters, skin side up and side by side directly on the oven grate. Broil for about 15 minutes - while spritze-basting about every 5 minutes. Turn the chicken quarters over and broil for about 15 minutes, spritze-basting as before. Turn the quarters over. Season them with salt and pepper. Broil until the skin flaps. Remove the quarters from the broiler. Allow them to rest for 5 minutes - before serving. Or, serve with a variation below.
Place mustard powder in a bowl. Add water, a Tbl at a time, to liquify flour to the desired consistency. Add the dark brown sugar and salt to taste. Stir or use the stick blendor to combine. Cover and refrigerate for 1 week before using.
When the chicken quarters are finished, slather the homemade mustard over the chicken then season. Broil until the mustard boils. Serve more of the sweet mustard on the side.
Pour peanut oil into a heavy, non-aluminum pan. Over low heat, bring the oil to 200 degress F, checking the temperature with a deep-fat thermometer. Remove the pan from the heat, stir in the cayenne powder and cool to room temperature. Pour into a container with a tightly fitting lid. Allow the oil to mature for a week. Warm the oil - again - and strain it through 2 paper coffee filters with a funnel into a clean glass jar.
When the chicken quarters are finished and seasoned, plate them then drizzle with the fiery oil.