Combine first 5 ingredients in a blender; process on high 1 minute. With blender on high, add oil in a slow, steady stream; process 1 minute. Scrape down sides. Add basil and thyme; process on low 30 seconds. Remove 1 cup sauce for basting and 1/2 cup sauce to serve with chicken; chill.
Place chicken in a large shallow dish or heavy-duty zip-top plastic bag. Pour remaining sauce over chicken; cover or seal, and chill 8 hours. Drain and blot excess sauce.
Grill chicken according to directions below. Brush 1 cup reserved sauce over chicken during the last 30 minutes of cooking time. (Discard leftover basting sauce.) Serve chicken with reserved 1/2 cup sauce. Garnish, if desired.
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: If you prefer chicken pieces, prepare a hot fire by piling charcoal on 1 side of grill, leaving other side empty. (For gas grills, light only 1 side.) Place cooking grate on grill. Arrange chicken over empty side (unlit side of gas grill), and grill, covered with grill lid, 1 hour or until chicken is done.