Total Time: 30 minute In 6- to 8-quart noncorrosive (stainless steel, enamel-coated or plastic) container or stockpot, mix water, kosher salt and brown sugar, stirring until salt and sugar are dissolved.
Stir in beer. Add chicken. Cover; refrigerate at least 8 hours but no longer than 24 hours.
Line 15x10-inch pan with sides with foil. Remove chicken from brine; rinse thoroughly under cool running water and pat dry with paper towels. Discard brine. Place chicken in pan. Refrigerate uncovered 1 hour to dry chicken skin. Meanwhile, in small bowl, mix all rub ingredients except oil; set aside.
Heat gas or charcoal grill for indirect cooking. Brush oil over chicken; sprinkle rub mixture over chicken. For two-burner gas grill, heat one burner to medium; place chicken on unheated side. For one-burner gas grill, place chicken on grill over low heat. For charcoal grill, move medium coals to edge of firebox; place chicken over drip pan. Cover grill; cook 15 minutes.
Turn chicken over; cover grill and cook 20 to 30 minutes longer, turning occasionally, until juice of chicken is clear when thickest piece is cut to bone (170°F for breasts; 180°F for thighs and drumsticks.
Oven Directions: Make chicken as directed through step 3. Heat oven to 425°F Brush oil over chicken; sprinkle rub mixture over chicken. Bake 25 to 35 minutes or until juice of chicken is clear when thickest piece is cut to bone (170°F for breasts; 180°F for thighs and drumsticks).
High Altitude (3500-6500 ft): In step 4, turn chicken over; cover grill and cook 25 to 35 minutes longer, turning occasionally.