Whisk buttermilk, brown sugar, cider vinegar, garlic, thyme, pepper, paprika, cumin, cayenne, and salt together in a bowl.

Trim wing flats from the chicken. Cut along both sides of the backbone from the tail to the neck; remove backbone. Make a 1/4-inch-deep cut through the piece of cartilage in the center. Open up both sides to reveal breastbone; lift out. Cut chicken in half through the center line.

Place both halves in a resealable zip-top bag and pour in the buttermilk marinade. Place the bag in a bowl to catch any leaks. Press air out and seal. Marinate in the fridge for 6 to 18 hours.

Preheat an outdoor grill for medium heat and lightly oil the grate. Remove chicken from marinade and transfer onto a paper towel-lined plate. Blot off excess marinade.

Place chicken on the grill cut-side down, skin-side up; cover and cook for 20 minutes.

Flip chicken over and continue grilling for 7 to 10 minutes. Flip once more and grill until no longer pink in the center, about 15 minutes more. An instant-read thermometer inserted into the thickest part of the thigh, near the bone, should read 165 degrees F (74 degrees C).