In a sauce pan - over medium heat, break down the roux in the water and the wine. Add the worcestershire, Tabasco and salt to taste. Bring the liquids to a boil. 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.