In a stockpot, heat 1 quart of water with 1 1/3 cups kosher salt and 2/3 cup packed light brown sugar, stirring to dissolve.

Remove from heat and add 6 quarts of cold water, 6 smashed garlic cloves, 2 teaspoons peppercorns, and half of the 20 sage and 20 thyme sprigs.

Let the brine cool to room temperature.

Discard the leg truss from the 14-16 lbs turkey.

Remove the neck, tail, and giblets from the turkey cavities and save for gravy if desired.

Pull off and discard lumps of fat.

Rinse the turkey inside and out.

Set the turkey in an oversized pot and add the cooled brine.

Chill, covered, for 8 to 12 hours.

Soak 4 fist-size applewood chunks in a bowl of water for at least 45 minutes, using a plate to keep them submerged.

Drain the turkey, rinse, and pat dry.

Set the turkey on a rack on a rimmed baking sheet to dry the skin slightly.

Chop the leaves from the remaining thyme.

Chop enough of the remaining sage to make 3 tablespoons.

In a food processor, whirl 1/2 cup unsalted butter (at room temperature), chopped herbs, and 1/4 cup finely chopped shallots until smooth.

Slide your fingers between the skin and flesh of the turkey to create pockets on the breast, back, and leg areas.

Using your hands, slide about 1 tablespoon of herb butter at a time under the skin in all the pockets.

Stuff the main turkey cavities with the remaining sage sprigs and skewer closed.

Tuck the wing tips under the turkey.

Insert a meat thermometer into the thickest part of the breast, ensuring it touches the bone.

Rub the turkey with 2 tablespoons olive oil.

Set up a charcoal grill for indirect low heat (250° to 300°F).

Light 40 charcoal briquets in a chimney starter.

Open the grill's bottom vents fully.

When the coals are spotted with ash (about 20 minutes), place an aluminum drip pan (at least as large as the turkey) in the middle of the firegrate.

Bank the coals on either side of the pan, piling them highest against the walls of the grill.

Fill the drip pan halfway with hot tap water.

Let the coals burn to low.

Drain 2 soaked wood chunks and put 1 over each mound of coals now and after 1 hour of cooking.

Put the cooking rack in place with its hinged flaps aligned over the coals.

Position the turkey on the cooking rack over the drip pan with the breast up.

Cover the grill with the lid and close the bottom vents and lid vents halfway.

Cook and smoke the turkey.

Whenever the heat dips below 250°F, add about 5 briquets to each pile of coals.

If the heat climbs over 300°F, close the vents, leaving at least a vent halfway open on the lid and at the bottom of the smoker to keep the fire going.

Add hot water to the drip pan as needed to keep it about half full.

Smoke until the thermometer registers 165° to 170°F, approximately 3 1/2 to 4 hours total.

Transfer the turkey to a platter and tent with foil.

Save the pan juices for gravy if desired.

Let the turkey rest in a warm place for 15 to 30 minutes, then carve.