In a large stockpot, brining bag or deep 5- to 6-qt. bowl, combine the brown sugar and 1/3 cup coarse salt. Add 2 cups very hot water and whisk until the sugar is dissolved. Stir in 6 cups cold water. Rinse the whole chicken inside and out with cold water; transfer it to the sugar-salt brine. If needed, place a heavy plate on top of the chicken to keep it submerged; let stand in the refrigerator or another cool place for 2 hours.
Meanwhile, place the wood chips in a medium bowl. Add 2 cups warm water and let stand for at least 30 minutes; drain.
Remove half of the grate from a gas grill; preheat to 400 degrees . Remove the chicken from the brine and rinse well inside and out with cold water. Open the beer can and discard or reserve half of the liquid. Insert the top of the can into the chicken's cavity and stand the chicken upright on a large baking sheet or cutting board. Carefully fold back the wings. Rub the chicken with the EVOO.
Transfer the wood chips to an aluminum tray or smoker box and place on the side of the grill without the grate. Lift the chicken off the baking sheet (with the beer can still inside) and place upright on the grate toward the back of the grill; cover the grill and maintain a temperature of 400 degrees to 425 degrees , increasing the heat to high under the wood chips and reducing to low under the chicken if necessary. Roast until the chicken is golden and crisp and the internal temperature of the thigh registers 160 degrees on a meat thermometer, about 1 hour.
Holding a kitchen towel in one hand and a sturdy metal spatula in the other, transfer the chicken to a clean baking sheet; let stand for 15 to 20 minutes. Remove the can from the cavity; discard. Carve the chicken or cut into pieces and serve.