Whisk together the first 6 ingredients in a large bowl until salt has dissolved.Add chicken pieces. Cover and refrigerate for 24 hours.
Remove chicken from marinade. Discard marinade.Blot chicken dry. Season with pepper.
Heat gas grill to medium (350 degrees to 450 degrees.) Keep burner turned off on one side of grill so you have an indirect heat area.
Place chicken skin side up on the indirect heat area of the grill. Cover and grill until a meat thermometer in the thickest portion registers 165 degrees F. This will take a little over an hour. Check both white meat and dark meat. The white meat will cook faster.
While chicken grills, whisk together the sauce ingredients.
When chicken has reaches 165 degrees, brush it with sauce and flip it over, placing it over direct heat for 2 minutes.Brush with more sauce and flip for another 2 minutes.
Serve chicken with the remaining sauce.
Notes
Note: If breasts are really large, cut them in half.I love to serve this chicken drizzled with a little blue cheese dressing. So good!