Sprinkle rub on all sides of brisket, rubbing onto the meat. If time permits, wrap brisket in plastic wrap and let cure in the refrigerator for at least 4 hours or overnight.
If using a smoker: Set up smoker according to manufacturer's instructions and bring temperature to 250 degrees. When ready to cook, place brisket fat side up in smoker. Add wood chips or chunks to the smoker every hour, following manufacturer's instructions.
If using a charcoal grill: Set up grill for indirect grilling, place a large drip pan in the center and preheat grill to 250 degrees. To maintain this low temperature, use only half as much charcoal as usual (a half chimney-full). When ready to cook, toss about 1 1/2 cups of wood chips or chunks on the coals. Place brisket on hot grate over drip pan, fat side up, and cover grill. Add fresh coals and more wood chips or chunks to each side of the grill every hour for the first 4 hours.
Smoke or grill brisket until a dark "bark" (outside crust) forms and internal temperature reaches 160 degrees, 7 to 9 hours. Then tightly wrap brisket in a couple of layers of aluminum foil, crimping edges to make a tight seal. Return brisket to smoker or grill and continue cooking until brisket is very tender—internal temperature should be 190 to 195 degrees, 3 to 4 hours longer. Cooking times are approximate and will vary depending on type of smoker or grill.
Remove wrapped brisket from smoker or grill and place it, still wrapped in foil, in a warm spot, such as an insulated cooler lined with newspapers or towels. Let rest for about 30 minutes. To serve, unwrap brisket and thinly slice. Spoon any juices over brisket and serve barbecue sauce on the side.
Serves 6 to 8