Preheat an outdoor barbecue for indirect medium heat.
Mix together dark brown soft sugar, sea salt, paprika, ground black pepper and cayenne pepper in a small bowl. Place two half full cans of beer on a baking tray. Spoon 1 teaspoon seasoning mix into each can. Be careful, this will make the beer foam up and out of the can.
Rub each chicken with 2 tablespoons vegetable oil. Rub the remaining seasoning mix over the entire chicken, inside and out. Fit each chicken over a can of beer with the legs on the bottom; keep upright.
Drain the wood chips and place them with the coals or in an tin on or under the barbecue grate as directed by the barbecue's manual. Place the two chickens, standing on their cans directly on the barbecue. Close the lid and barbecue the chicken until no longer pink at the bone and the juices run clear, about 1 hour 30 minutes. An instant-read thermometer inserted into the thickest part of the thigh, near the bone should read 82 degrees C. Remove the chickens from the barbecue and discard the beer cans. Cover the chicken with a doubled sheet of foil and allow to rest in a warm area for 10 minutes before slicing.