Step 1: Make the Cognac Herb Butter: In a mixing bowl, combine the butter and herbs and beat with a wooden spoon until light and fluffy. Beat in the cognac and lemon zest; add salt and pepper to taste. (You can make the Cognac Herb Butter ahead and refrigerate or freeze it, but let it come to room temperature before using it.) Set aside.
Step 2: Make the injector sauce: Melt the butter in a saucepan over medium heat. Stir in the chicken broth and cognac and let cool.
Step 3: Remove the giblets, if any, and any large lumps of fat from the main cavity of the turkey. (Reserve the giblets for another use.)
Step 4: Starting at the top of the neck cavity, tunnel your finger under the skin. Start with one finger, then two. Gently loosen the skin from the meat, taking care not to tear it. Gradually work your whole hand under the skin, loosening it from the breast meat, then the thighs.
Step 5: Spoon 3/4 of the Cognac Herb Butter under the skin and use your hands to distribute it as evenly as possible. (Massaging the outside of the bird will help distribute the butter.) Position the truffle slices or sage leaves under the skin on each half breast and thigh. Fold the wings behind the back. Reserve the remaining butter to use for basting.
Step 6: Truss the turkey with butcher’s string. Inject it with injector sauce.
Step 7: Set up your grill for indirect grilling and preheat to 325 to 350 degrees. For a smoke flavor, add wood chips or chunks to the coals. (If using a gas grill, place the chips in the smoker box or make a foil smoking pouch.)
Step 8: Smoke-roast the turkey for 1 hour. Baste the turkey with some of the reserved Cognac Herb Butter. Add more chips to the coals if using. Continue smoke-roasting the bird for 1-1/2 to 2 hours more, basting every 30 minutes, until the skin is a dark golden brown and the meat is cooked through. The internal temperature of the thickest part of the thigh when read on an instant-read meat thermometer should be 160 degrees. (The temperature will rise as the turkey rests.) If the skin starts to brown too much, lay a sheet of aluminum foil over the bird.
Step 9: Carefully transfer the turkey to a cutting board or platter and baste with the Cognac Herb Butter one final time. Let rest for 15 to 20 minutes before carving.
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Steven Raichlen’s Project Smoke is a production of Maryland Public Television, Barbacoa, Inc., and Resolution Pictures. © 2017 Barbacoa, Inc.