Light a grill, preferably charcoal. Rub the chicken with the oil and season with the thyme, red pepper, salt and black pepper. When the fire is medium hot, arrange the chicken, skin side down, on the grill. Cover and cook, rotating from time to time, so the fat is rendered, about 10 minutes. Turn the chicken, cover and continue cooking until almost cooked through, about 15 minutes longer. Uncover and generously spoon the True Texas Barbecue Sauce all over the skin. Turn the chicken side down and grill, basting and turning so it doesn't burn, until the pieces are nicely glazed and slightly charred, about 10 minutes. Serve hot off the grill.