When roasting a whole chicken on the grill, begin with a plump roaster or broiler-fryer. You can also roast capons on the grill, which are fat, compact male birds that are castrated at an early age to promote plumpness, tenderness, and sweet flavor.
Combine the beer, Worcestershire sauce, onion, mustard, salt, and paprika in a small bowl, stirring well.
Put the chickens, breast side down, in a shallow glass or ceramic dish and pour the marinade over them. Cover and refrigerate for 4 hours or overnight.
Prepare a charcoal or gas grill, arranging the coals for indirect cooking or lighting one side of the grill. Lightly spray the grill rack with vegetable oil cooking spray. The coals should be moderately hot.
Lift the chicken from the marinade. Transfer the marinade to a small saucepan. Place the chickens over the cooler part of the grill, cover, and cook for 1 hour and 20 minutes to 1 hour and 30 minutes, or until the juices run clear when the thigh meat is pierced with a fork or sharp knife. An instant-read thermometer inserted in the thickest part of the thigh should register 180°F. Don't let the thermometer touch the bone. Add fresh coals to the fire as necessary to maintain a moderate, constant heat. If using a gas grill, adjust the burner furthest from the chicken to medium. Turn the chicken with tongs 3 or 4 times during grilling to brown evenly on all sides.
Meanwhile, bring the marinade to a boil over medium-high heat. Reduce the heat and simmer for about 20 minutes until reduced to about 1 cup. Add the molasses, stir, and cook for 5 minutes longer. Remove 1/4 cup of the marinade and set aside the rest.
During the final 5 minutes of grilling, brush the chicken with the 1/4 cup of marinade, turning the chicken to crisp the skin on all sides. Let the chickens sit for at least 10 minutes for the juices to collect. Serve the remaining marinade with the chicken.