Remove giblets from inside of turkey and rinse bird with fresh water.
Brine chicken for 4-24 hours in a wet brine or simply rinse with fresh water. Add brine ingredients in a large pot and bring to a boil to dissolve salt & sugar. Remove from heat.
Allow brine to cool COMPLETELY before adding & submerging turkey into liquid. Once finished, rinse with cool fresh water.
Dry turkey with paper towels and tie the legs together with butcher twine or leave on plastic leg tie that comes with most turkeys.
Add dry seasoning ingredients to the softened butter and mix well. Rub seasoned butter all over the turkey. Make sure to get both sides of the bird and in between the skin and breast meat.
Lay turkey directly on grill rack breast side up and smoke for 3 hours. Spritz bird with chicken broth every 45 minutes beginning after the first 1 hour of smoking.
Increase temperature to 350°F and cook until the internal temperature is 165°F when checked with an instant read thermometer, about another 1 hour.
Let rest 10-15 minutes and serve.
Notes
Tie turkey legs together to prevent the breast meat from cooking faster than the legs and thighs. If turkey came with plastic leg holders, leave on and use instead of twine.