Make brine: In a large pot, heat 1 qt. water with salt and sugar, stirring to dissolve. Remove from heat, add 5 qts. cold water, garlic, cinnamon, and chiles, and let cool to room temperature.
Remove leg truss from turkey; discard. Remove neck, tail, and giblets. Discard tail; save the rest, chilled, for Mole Gravy. Pull off and discard lumps of fat. Rinse bird inside and out. Lower turkey into brine (see tips below). Chill, covered, 12 hours.
Thanksgiving Day: Prepare grill for indirect medium-low heat (about 325°). For charcoal: Light 40 briquets on firegrate. When coals are spotted with ash, 20 minutes, bank evenly on opposite sides of firegrate and let burn to medium-low. Set a metal or foil drip pan (about 9 by 15 in. and 2 1/2 in. deep) between coal mounds and fill with 1/2 in. warm water. To each mound of coals, add 5 briquets now and every 30 minutes while cooking. Oil cooking grate and set in place. For gas: Set the metal or foil drip pan on center burner(s) and fill with 1/2 in. warm water. Turn burners except for one(s) under drip pan to high, close lid, and heat 10 minutes. Reduce heat for active burners to 325°. Oil cooking grate and set in place.
Remove turkey from brine, rinse, and pat dry. Discard brine. Tuck wing tips under turkey.
Make rub: Combine ground chiles and oregano. Rub turkey all over with oil, then sprinkle evenly inside and out with seasoning.
Set turkey, breast up, on cooking grate over drip pan. Grill, covered, until a meat thermometer inserted straight down through thickest part of breast to the bone registers 160°, 2 to 3 hours; as turkey cooks, use foil to tent any areas that start to get too dark.
Tip turkey so that cavity juices run into drip pan, holding turkey with oven mitts. Transfer turkey to a cutting board; save pan juices for gravy. Tent turkey with foil and let rest in a warm place 15 to 30 minutes while you make gravy.
Carve and serve with gravy.
Brining. It's the secret to a moist bird. Brine the turkey, breast down, in an oversize pot such as boiling-water canner (if it fits in the fridge). Or use an ice chest. Bag turkey in a turkey brining bag (available in grocery stores and from surlatable.com). Put bagged bird in chest, pour brine over turkey, and seal bag tightly. Add ice to cover.
Tying. For the most even cooking, leave turkey legs loose. Or if you prefer a tidy look, tie them.
Garnishing. Add color with herbs, a banana leaf, and fruit like kumquats.
Note: Nutritional analysis is per 1/4 lb. serving of white and dark meat with skin.