To make Bobby Flay's Cajun Brined Turkey, combine paprika, oregano, thyme, cayenne, and chile de arbol in a bowl, and then add chile, garlic and onion powders. Add salt, sugar and half of the spice mixture to 2 gallons of water. Submerge the turkey, along with the onion, garlic and bay leaves, and refrigerate for 12 hours. Rinse and allow to dry for 24 hours. Brush with oil and spices and cook using a Big Green Egg or Caja China.
To cook in a Big Green Egg, fill the cooker's chimney with 5 quarts of charcoal briquettes. Burn the charcoal until slightly ashy, and then put the coals on the grill. Place the ceramic plate on top of the charcoal, and then put the grill rack on top of the grate. Cover, and allow to heat until it reaches 350 F. Add the turkey, and cook for two hours or until its internal temperature is 165 F.
To cook in the Caja China, place the turkey on the roasting pan inside the cooker. Close the top, and add charcoal. Cook the turkey for 30 minutes, and then flip it and cook for one additional hour or until a thermometer indicates it has reached 165 degrees.