Heat a medium-sized Dutch oven over a medium-high flame. Pour the peanut oil into the pan. Allow the oil to come up to temperature, sprinkle the flour into the oil, while whisking with a long-handled, wire whip. From this point on, whisk the roux continuously. Allow the roux to acquire a reddish brown color. The caramelization process should take 6-8 minutes. Turn the flame down, then - carefully - pour the wine and water into the pan. Stand back to allow the erupting volcano of liquid and hot roux to subside. Combine the liquid and roux with fast whisking. Add the worcestershire, Tabasco and salt to taste. Simmer the sauce over a moderate flame.
Inspect the progress of the chicken. The halves should have picked up color and crust. Turn the halves over. Continue grilling and turning the chicken until the juices run clear, about 45 minutes in total. Remove the chicken from the grill and season - aggressively - with salt and pepper. Before saucing, allow the chicken to rest 5-8 minutes.
Check and stir the sauce periodically. It is finished when it coats the back of the spoon. Do not over-reduce the sauce, as it will become too potent. Pour the sauce over both halves of the chicken. Serve with rolls, rice or biscuits.