Cut off pointed end of garlic bulb; place garlic on a piece of aluminum foil, and drizzle with olive oil. Fold foil to seal.
Bake at 425° for 30 minutes; cool. Squeeze pulp from garlic cloves.
Combine garlic pulp, ketchup, and next 12 ingredients in a large saucepan. Bring mixture to a boil; reduce heat, and simmer 20 minutes. Cool.
Pour mixture in a blender; process until smooth, stopping once to scrape down sides.
Reserve 1 cup barbecue sauce for basting chicken the last 30 minutes of grilling time. (Discard any leftover basting sauce.)
Grill chicken according to directions below. Serve chicken with remaining sauce.
For Gas Grills:
Indirect Cooking: Place 2 cups hickory, mesquite, or other wood chips in the center of a large square of heavy-duty aluminum foil; fold into a rectangle, and seal. Punch holes in top of packet. Preheat one side of grill, leaving center empty, for 20 minutes. Place packet on cooking grate over unlit side. Grill, covered with grill lid, 2 hours and 15 minutes or until done (170°). Baste as directed.
Direct Cooking: Preheat grill over low heat, under 300°, for 20 minutes. Place chicken, skin side up, on cooking grate. Grill, covered with grill lid, over low heat 1 hour and 15 minutes or until done. (Don't turn chicken.) Baste as directed.
For Charcoal Grills:
Indirect Cooking: Soak 2 cups hickory wood chips in cold water for 30 minutes; drain. (Wood chips for the gas grill are not soaked in water because they're encased in foil and placed on the cooking grate.) Prepare fire by piling charcoal on each side of grill, leaving center empty. Let charcoal burn for 30 minutes, or until flames disappear and coals turn white. Sprinkle chips over hot coals. Arrange chicken, skin side up, on cooking grate in center of grill (not directly over coals). Cook covered with grill lid, for 50 minutes to 1 hour or until done. (Don't turn chicken.) Baste as directed.
Note: For these methods, we prefer to use a 3-pound chicken whole chicken cut into quarters, which tend to cook more evenly. If using chicken pieces, remove drumsticks and wings from the grill a little earlier to keep them from burning.