Two days before cooking: Combine the Spanish paprika, New Mexican chile powder, garlic powder, onion powder, oregano, thyme, chile de arbol and cayenne in a small bowl. Remove 1/4 cup for another use, such as the Eggplant Casserole with Red Pepper Pesto and Cajun Breadcrumbs.

Dissolve the salt and sugar in 2 gallons cold water in a large (at least 16-quart) stockpot or a clean bucket. Whisk in half of the remaining spice rub, the garlic, bay leaves and onion. Add the turkey, cover and refrigerate or set in a very cold place (that stays somewhere between 32 to 40 degrees F) for at least 12 hours and up to 24 hours. Rinse well under cold water and pat dry to remove the salt. Let air-dry in the refrigerator for 24 hours.

In The Big Green Egg: Light a large chimney filled with charcoal briquettes (about 5 quarts) and allow to burn until all the charcoal is covered with a layer of fine gray ash. Empty the coals into the grill. Top with some of the soaked wood chips if using. Put in the ceramic plate, then the grill rack. Put the cover on and let the grate heat up.

Brush the turkey with oil and rub the entire turkey (including the cavity) with the remaining spice rub. Tie the legs together. Adjust the grill vents to get and maintain an internal temperature of 350 degrees F. Roast the turkey for 2 hours to 2 hours and 15 minutes, until a meat thermometer inserted in the thigh reads 165 degrees F. Remove from the grill, tent with foil and let rest for 30 minutes before carving.

In the Caja China: Prepare the charcoal briquettes according to the instructions that come with the box. Put the roasting pan with its rack (comes with the box) inside, and put the turkey breast-side down on the rack. Put on the top and fill with charcoal. After 30 minutes, flip the turkey breast-side up and continue cooking for about 1 more hour, until a meat thermometer inserted in the thigh reads 165 degrees F. Add charcoal as necessary according to the box instructions. Remove from the box, tent with foil and let rest for 30 minutes before carving.