Preparation

• Soak the mesquite chips in water for at least 30 minutes.
• In a small bowl combine the sugar, smoked paprika, cumin, both chile powders, oregano, and 1-1/2 tsp. salt.
• Pat the chicken dry with paper towels. Using your fingers, gently separate the skin from the meat, taking care not to tear the skin. Spread half of the rub under the skin, massaging it directly onto the meat. Massage the remaining rub all over the chicken skin.
• Open the can of beer and pour out (or drink) about two-thirds. Using a church-key-style can opener, make two more holes in the top of the can. Place the beer can on a solid surface and lower the chicken cavity over the can. Rub the chicken all over with the oil. Let sit at room temperature while you prepare a gas or charcoal grill fire for indirect cooking over medium heat (350°F to 375°F).
• Drain and scatter the wood chips directly over the coals or put them in the smoker box of a gas grill, following manufacturer’s instructions. If using a gas grill without a smoker box, wrap the chips in a foil packet, pierce it in several spots, and lay the packet over a lit burner.
• When the chips begin to smoke, transfer the chicken on the can to the cooler side of the grill, balancing it on its two legs and the can like a tripod. Grill with the lid closed until an instant-read thermometer inserted in the thickest part of the breast (not touching the bone) reaches 165°F, 1-1/4 to 1-1/2 hours. (Halfway through cooking, turn the chicken 180 degrees to ensure even cooking. Cover the neck and wings with foil if they begin to brown too quickly.)
• Remove the chicken from the grill and let rest for 10 to 15 minutes. Carefully lift the chicken from the beer can, carve, and serve.