1. In a large bowl, mix buttermilk, shallots, garlic, salt, sugar, cumin, and pepper.
2. Rinse chicken thighs and drumsticks and pat dry. Trim off excess fat. Submerge chicken pieces in buttermilk brine. Cover and chill for at least 4 hours, or up to 1 day (see notes).
3. Lift chicken from brine; discard brine. Wipe excess from chicken with paper towels.
4. Lay chicken pieces on a barbecue grill over medium coals or medium heat on a gas grill (you can hold your hand at grill level only 4 to 5 seconds); close lid on gas grill. Cook, turning frequently, until browned on both sides and no longer pink at the bone (cut to test), 20 to 30 minutes. Serve hot or cold.
Brine the chicken up to 1 day before grilling; grill up to 1 day before serving (chill airtight). Transport in brine when grilling on-site. Keep the meat well chilled in an ice chest until ready to serve or grill.