Prepare for brining. Start brining the turkey 24 hours before you plan to start smoking it. Make sure that your cooler is clean and dry, and that your drain plug is inserted properly. Or clear space in your refrigerator if you plan on brining the turkey in the fridge. Spatchcock the thawed turkey.
Make the brine. In a large pot, add these ingredients: water, black peppercorns, bay leaves, salt, brown sugar, white sugar, lemon wedges, and halved garlic. Heat over medium heat until boiling, stirring often for the sugars to dissolve. Then, remove the pot from the heat and pour in the full jar of Cajun marinade. Stir to combine. Allow the brine mixture to cool for about 30 minutes.
Brine the turkey. Add a 10 pound bag of ice to the bottom of a cooler. Stir in the Cajun brine. Lay your turkey in the cooler, breast side down. Stir the ice and brine around a bit so that the turkey is completely submerged in the brine. Close the lid of the cooler and brine turkey for 18-24 hours. *See notes.
Preheat the smoker. Preheat the grill or smoker to about 300℉ using your choice of wood or pellets.
Dry and season the turkey. Remove the turkey from the brine and transfer to a wire cooling rack on top of a sheet tray. Pat the turkey dry with paper towels. Liberally season all sides of the turkey with the Cajun seasoning and the citrus rub.
Smoke the turkey. Insert wireless meat thermometer into the thickest part of the breast to monitor the internal temperature. Put the turkey on the grill with the breast side on the opposite side of the heat source. Close the lid and smoke turkey at 300℉ until internal temp in the thickest part of the breast and the innermost area of the thigh and wing reaches 162-165℉. (USDA recommends cooking to 165℉. A 14 pound turkey takes roughly 4½ – 5 hours to smoke at 300℉, but be sure to check internal temperatures for doneness.)
Rest and serve. Remove turkey from the smoker and transfer to a serving platter or cutting board. Allow turkey to rest for 30 minutes before slicing and serving.