Begin preparing the turkey and the stock for the gravy one day before serving. Remove the neck and giblets from the turkey (discard the kidneys and liver) and coarsely chop them. Set aside. Using paper towels, pat the turkey as dry as possible, inside and out. Set the turkey, breast side up, on a rack set inside a roasting pan. Refrigerate, uncovered, for 12 to 24 hours.
To make the stock for the gravy, place a 4- to 5-quart saucepan over medium-high heat and add the vegetable oil and 1 tablespoon of the butter. Sauté the neck and giblets with the carrot, onion, and celery until all the ingredients are browned but not scorched, 10 to 12 minutes, stirring frequently. Add the salt, pepper, garlic, thyme, wine, and sherry and stir to scrape up all the delicious bits that have stuck to the pan. Add the broth, bring it to a boil, and then reduce the heat to maintain a low simmer for 1½ hours, uncovered. Every 20 to 30 minutes, remove any grease that has risen to the top with a large, shallow spoon. Strain the stock through a fine-mesh sieve into a large measuring cup, pressing down on the solids to extract all the liquid. If there is less than 4 cups, add more chicken broth to make up the difference. Pour the stock into a smaller saucepan and place it over medium-high heat, simmering it briskly until it is reduced to about 2 cups, 20 to 25 minutes. Cool, cover, reserve, and refrigerate the stock overnight or until ready to make the gravy. Skim off any congealed fat.
Remove the turkey from the refrigerator, loosely cover with paper towels, and allow it to stand at room temperature for 1 hour before smoking.
Soak the wood chips in water for at least 30 minutes.
Prepare a half circle or crescent-shaped fire for medium-low heat (300° to 350°F).
Set the turkey, breast side down, on a work surface and rub 1 tablespoon of the olive oil and 2 teaspoons of the salt into the skin, concentrating the salt over the thighs. Turn the bird over and repeat this step, concentrating most of the salt over the breast and getting some inside both the neck and body cavities. Place the turkey breast side up, and fill the cavity with the lemon, thyme sprigs, and garlic. Transfer the turkey to a roasting rack and place the turkey, on its rack, in the roasting pan.
In a large skillet, melt 1 cup of the butter, and then stir in the sherry. Remove the mixture from the heat and allow it to cool for 5 to 10 minutes. Cut a 2½-foot length of cheesecloth, and open it out fully into a single layer. Fold this over, crosswise, to form a doubled sheet 15 inches long. Dunk the cheesecloth blanket into the butter-sherry mixture, holding it by the ends, and saturating it completely. Lift the blanket and let the excess butter mixture drip back into the skillet for a moment, then drape the cheesecloth blanket over the turkey, stretching it taut, tucking it in snugly around the legs and thighs, and keeping the wing tips in close to the breast.
Place the turkey in the pan on the side of the cooking grate opposite the coals, with the legs facing the coals. It may be necessary to bend in the corners of the pan in order to close the lid of the grill. Drain and add 2 cups of the wood chips to the charcoal. Cook the turkey over indirect medium-low heat, with the lid closed as much as possible, brushing the cheesecloth with the butter-sherry mixture every 30 minutes. Briefly rewarm the basting sauce if the butter has begun to harden. Adjust the vents to keep the temperature within the 300° to 350°F range. After the first hour of cooking time, drain and add the remaining wood chips. When the legs are golden brown, after 30 to 40 minutes, rotate the bird so the breast faces the coals, and continue to rotate it with each basting, so that one side, and then the opposite side, also faces the fire. While the turkey is cooking, add additional lit coals as needed to maintain the temperature between 300° and 350°F. Cover the drumsticks (and the breast, if necessary) with a strip of aluminum foil if they are browning too fast. Plan on 2½ to 3½ hours of cooking time, depending on the weight of the turkey. Drain and add the remaining wood chips after 1½ hours of cooking time.
Twenty to 30 minutes before the turkey is done (when the internal temperature of the breast reaches 140° to 150°F), carefully remove and discard the cheesecloth blanket and baste the turkey directly on its skin. This will help achieve a uniform dark golden color. Continue cooking until the internal temperature of the thigh (not touching the bone) reaches 170° to 175°F, rotating the bird as necessary for even browning. Remove from the grill, cover loosely with foil, and let rest for about 30 minutes (the internal temperature will rise 5 to 10 degrees during this time).
While the turkey is resting, finish the gravy. In a medium saucepan over medium-low heat, melt the remaining ¼ cup butter and stir in the flour to make a smooth paste. Cook until it is bubbly and turns light brown, 3 to 4 minutes, whisking constantly. Add the reserved stock to the saucepan and bring it to a simmer. Cook gently, uncovered, for a few minutes until the gravy has thickened, whisking constantly. Stir in the cream, return to a brisk simmer, and cook until it looks glossy, 1 to 2 minutes longer. Stir in the vinegar. Remove the gravy from the heat and season with salt and pepper. Keep it warm until serving time. Use up to 1 cup additional chicken broth to thin the gravy to your desired consistency as needed.
Carve the turkey and serve with the gravy.
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