1. Prepare grill for indirect grilling, heating one side to medium.
2. Combine first 7 ingredients and 1/2 teaspoon salt in a small bowl.
3. Open both root beer cans; pour 18 ounces into a small saucepan. Set remaining root beer aside (in the can). Bring 18 ounces root beer to a boil. Cook until reduced to 1/3 cup (about 20 minutes). Remove from heat. Add remaining 1/4 teaspoon salt, butter, and vinegar, stirring until smooth.
4. Rub paprika mixture evenly over chicken. Holding chicken upright with the cavity facing down, insert reserved opened root beer can into cavity. Place chicken on unheated side of grill. Spread legs out to form a tripod to support the chicken. Cover and grill for 1 hour and 30 minutes or until a meat thermometer inserted into meaty portion of thigh registers 160°, basting chicken every 20 minutes with sauce.
5. Lift chicken slightly using tongs; place spatula under can. Carefully remove chicken and can from grill; place on a cutting board. Let stand 10 minutes. Gently lift chicken using tongs or insulated rubber gloves; carefully twist can and remove from cavity. Discard can. Carve chicken.