For the chicken, put the chicken halves in a very large bowl or deep pot and cover with water. Sprinkle 3 tablespoons of the salt in the water, cover and refrigerate the chickens for 6 hours or overnight.
For the BBQ sauce, mix together the vinegar, remaining tablespoon salt, peanut oil, hot sauce, black pepper, 1/4 cup water and cayenne in a saucepan. Bring this mixture to a boil, stir well and remove from the heat.
Drain the chicken, pat dry and place the halves on the grill, skin side up. Baste with the sauce and cook for 30 minutes. Turn the chickens skin side down and baste the top with sauce. Continue to grill the chickens for an additional 30 minutes, turning and basting the chicken every 10 minutes. Check for doneness by twisting a drumstick; it should move easily.
Cook's Note: If you cannot find small chickens, use larger ones (3 to 3 1/2 pounds) and quarter them.
Recipe adapted from Georgia Cooking in an Oklahoma Kitchen by Trisha Yearwood (c) Clarkson Potter 2008