Make brine: In a stockpot, heat 1 qt. water with salt and brown sugar, stirring to dissolve. Remove from heat, add 6 qts. cold water, the garlic, peppercorns, and half of the sage and thyme. Let cool to room temperature.
Discard leg truss from turkey. Remove neck, tail, and giblets from cavities and save for gravy if you like. Pull off and discard lumps of fat. Rinse bird inside and out. Set turkey in an oversized pot, such as a boiling water canner*, and add brine. Chill, covered, 8 to 12 hours.
Soak wood chunks in a bowl of water until used (at least 45 minutes), with a plate on top to submerge them. Meanwhile, drain turkey, rinse, and pat dry. Set on a rack on a rimmed baking sheet to dry skin a bit.
Chop leaves from remaining thyme. Chop enough of remaining sage to make 3 tbsp. In a food processor, whirl butter, chopped herbs, and shallots until smooth.
Slide your fingers between skin and flesh of turkey, turning bird to make pockets of space on the breast, back, and leg areas. Using your hands, slide about 1 tbsp. herb butter at a time under skin in all the pockets. Stuff main turkey cavities with remaining sage sprigs and skewer closed. Tuck wing tips under turkey. Insert a meat thermometer into thickest part of breast, touching the bone. Rub turkey with oil.
Set up a charcoal grill for indirect low heat (250° to 300°): Light 40 briquets in a chimney starter. Open grill's bottom vents fully. When coals are spotted with ash, about 20 minutes, put an aluminum drip pan (at least as large as turkey) in middle of firegrate. Bank coals on either side of pan, piling them highest against walls of grill. Fill drip pan halfway with hot tap water. Let 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 cooking rack in place with its hinged flaps aligned over coals. Position turkey on cooking rack over drip pan with breast up. Cover with lid and close bottom vents and lid vents halfway.
Cook and smoke turkey. Whenever heat dips below 250°, add about 5 briquets to each pile of coals. If the heat climbs over 300°, close vents, leaving at least a vent halfway open on lid and at bottom of smoker to keep fire going. Add hot water to drip pan as needed to keep it about half full. Smoke until thermometer registers 165° to 170°, 3 1/2 to 4 hours total.
Transfer turkey to a platter and tent with foil. Save pan juices for gravy if you like. Let turkey rest in a warm place 15 to 30 minutes, then carve.
* Find applewood chunks with grilling supplies. If pot won't fit in fridge, use an ice chest: Set bird in a turkey brining bag (buy in grocery stores and at williams-sonoma.com). Put bag in chest, pour brine over turkey, and seal bag tightly; add ice to cover.
Make ahead: Wrap drained, brined turkey airtight and chill up to 1 day before smoking.