Make the brine: Combine 1 gallon of water, kosher salt, brown sugar, peppercorns, bay leaves, garlic, thyme, and rosemary in a large stockpot. Bring to a boil, stirring until salt and sugar dissolve completely. Remove from heat.
Cool the brine: Add 1 gallon of ice water to the hot brine to bring the temperature down quickly. The brine must be completely cold before adding the turkey.
Brine the turkey: Place the turkey in a food-grade container and pour the cooled brine over it until fully submerged. Refrigerate for 12-24 hours, keeping the temperature below 40°F.
Remove and dry: Take the turkey out of the brine. Rinse briefly under cold running water and pat completely dry inside and out with paper towels. Discard the brine.
Make the injection: Melt the unsalted butter and combine with chicken broth, garlic powder, onion powder, smoked paprika, cayenne, and black pepper. Strain through a fine mesh sieve to remove any solids. Keep warm so the butter stays liquid.
Inject the turkey: Using a meat injector with a large-bore needle, inject the butter mixture in a grid pattern. Hit 3-4 sites on each side of the breast, 2-3 in each thigh, and 2-3 in each drumstick. Depress the plunger slowly while pulling the needle back to distribute the liquid evenly.
Season the exterior: Apply your favorite turkey rub or a simple blend of black pepper, garlic powder, and paprika to the entire surface of the turkey.
Cook the turkey: Smoke or roast at 275-325°F until the thickest part of the thigh registers 165°F on an instant-read meat thermometer. A 12-15 pound turkey typically takes 3-4 hours at this temperature range.
Rest and carve: Remove the turkey from the smoker or oven and let it rest for 20-30 minutes before carving. This allows the juices to redistribute throughout the meat for the best texture and flavor.