About 14 hours before grilling, in a very large bowl whisk all of the brine ingredients, except the ice water, until the salt and sugar are dissolved. Whisk in the ice water. The brine should be very cold.

Remove the giblets, neck, and lumps of fat from the turkey and reserve (discard the liver). Place in a bowl, cover, and refrigerate until ready to grill. If the turkey has a plastic truss or a pop-up timer, remove and discard it. Place the turkey inside a sturdy plastic bag and put it in a 10-quart or larger stockpot. Pour enough of the brine into the bag to cover the turkey as much as possible when the bag is closed and tightly tied. Discard any extra brine. Seal the bag. Refrigerate the turkey for at least 12 hours and no longer than 14 hours.

Remove the turkey from the bag and discard the brine. Rinse the turkey under cold water and pat it dry inside and outside with paper towels. Put one-third of the chopped onions and all of the thyme sprigs into the body cavity. Tuck the wing tips behind the turkey’s back and loosely tie the drumsticks together with butcher’s twine. Brush the turkey all over with the melted butter. Place one large disposable foil pan inside of the other to create a single pan of double thickness. Put the remaining onions into the pan and place the turkey, breast side down, on top of the onions. Allow the turkey to stand at room temperature for 1 hour before grilling.

Soak the wood chips in water for at least 30 minutes. Prepare the grill for indirect cooking over medium-low heat (about (350°F ).

Place a large foil pan on top of the Flavorizer bars, below the cooking grate. Fill it halfway with room temperature water. Prepare the grill for cooking over medium-low heat (350°F) .

Soak the wood chips in water for at least 30 minutes. Prepare the grill for indirect cooking over medium-low heat (about (300° to 350°F ).

Place the reserved giblets, neck, and lumps of fat in the roasting pan and pour in the chicken broth. Drain and add two handfuls of the wood chips to the charcoal or to the smoker box of a gas grill, following manufacturer’s instructions, and close the lid. When the wood begins to smoke, cook the turkey over indirect medium-low heat, with the lid closed, for 1 hour, keeping the grill’s temperature as close to 350°F as possible. If using a charcoal grill, replenish the charcoal as needed to maintain a steady temperature, adding 8 to 10 unlit briquettes every 45 minutes to 1 hour. Leave the lid off the grill for about 5 minutes to help the new briquettes light.

Place the reserved giblets, neck, and lumps of fat in the roasting pan and pour in the chicken broth. Drain and add two handfuls of the wood chips to the smoker box of a gas grill, following manufacturers instructions, and close the lid. When the wood begins to smoke, cook the turkey over indirect medium-low heat, with the lid closed, for 1 hour, keeping the grills temperature as close to 350°F as possible.

Place the reserved giblets, neck, and lumps of fat in the roasting pan and pour in the chicken broth. Drain and add two handfuls of the wood chips to the charcoal, following manufacturers instructions, and close the lid. When the wood begins to smoke, cook the turkey over indirect medium-low heat, with the lid closed, for 1 hour, keeping the grills temperature as close to 350°F as possible. Replenish the charcoal as needed to maintain a steady temperature, adding 8 to 10 unlit briquettes every 45 minutes to 1 hour. Leave the lid off the grill for about 5 minutes to help the new briquettes light.

Place the reserved giblets, neck, and lumps of fat in the roasting pan and pour in the chicken broth. Cook the turkey, above foil pan, over medium-low heat, with the lid closed, for 1 hour.

After 1 hour, turn the turkey over so that the breast faces up. Drain and add two handfuls of wood chips to the charcoal or to the smoker box. Cook the turkey for another 45 minutes, and then drain and add the remaining two handfuls of wood chips to the charcoal or to the smoker box. Continue cooking the turkey, with the lid closed, until an instant-read thermometer inserted into the thickest part of the thigh (not touching the bone), registers 170ºF , 45 minutes to 1¼ hours more.

After 1 hour, turn the turkey over foil pan, so that the breast faces up. Cook the turkey for another 45 minutes. Continue cooking the turkey, with the lid closed, until an instant-read thermometer inserted into the thickest part of the breast (not touching the bone), registers 165°F , 45 minutes to 1¼ hours more.

After 1 hour, turn the turkey over so that the breast faces up. Drain and add two handfuls of wood chips to the charcoal. Cook the turkey for another 45 minutes, and then drain and add the remaining two handfuls of wood chips to the charcoal. Continue cooking the turkey, with the lid closed, until an instant-read thermometer inserted into the thickest part of the breast (not touching the bone), registers 165°F , 45 minutes to 1¼ hours more.

After 1 hour, turn the turkey over so that the breast faces up. Drain and add two handfuls of wood chips to the smoker box. Cook the turkey for another 45 minutes, and then drain and add the remaining two handfuls of wood chips to the smoker box. Continue cooking the turkey, with the lid closed, until an instant-read thermometer inserted into the thickest part of the breast (not touching the bone), registers 165°F , 45 minutes to 1¼ hours more.

Remove the pan with the turkey from the grill. Tilt the turkey so the juices run out of the body cavity and into the pan. Transfer the turkey to a cutting board and let rest for 20 to 30 minutes (the internal temperature will rise a few degrees during this time). Save the pan juices to make the gravy.

Strain the pan juices into a large gravy separator. Let stand until the fat rises to the surface, about 3 minutes. Pour the pan juices into a large glass measuring cup, reserving the fat. If necessary, add chicken broth to the pan juices to measure 1 quart of liquid.

Measure the reserved fat. If necessary, add melted butter to make (½ cup) . In a medium saucepan heat the fat (and butter) over medium heat. Whisk in the flour and let it bubble for 1 minute, whisking constantly. Whisk in the pan juices and the lager. Bring the gravy to a simmer, whisking frequently. Reduce the heat to medium-low, stir in the thyme, and simmer until slightly thickened, 3 to 5 minutes. Remove from the heat and season with salt and pepper, if desired (the salt and pepper in the brine may have provided enough seasoning already).

Carve the turkey and serve with the warm gravy.