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 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 oversize 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 ready to use (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.
Ignite fire: Referring to manual, open vents and take smoker apart. On firegrate, fill charcoal chamber with charcoal so it's slightly mounded. Distribute paraffin cubes among briquets and ignite. Let burn until coals are just coated with ash, 25 to 30 minutes. Set water pan, then middle section of smoker in place. Protecting hands, pour hot tap water into water pan until it's nearly full.
Put top cooking rack in place and position turkey with breast up. Cover with lid and close vents halfway.
Using tongs, add 2 soaked wood chunks through smoker door to coals now and after 1 hour; for a 5-in. smoker, add last 2 chunks an hour later. Check water levels and temp every so often, keeping the smoker mostly closed so heat and smoke don't escape. Add hot water as needed to keep the pan about half full. The smoker's temp should read 200° to 250°; if needed, raise or lower by adjusting vents (leave a vent at least half-open on lid and at bottom of smoker to keep fire going). Smoke until meat thermometer registers 165° to 170°, 4 1/2 to 5 hours total.
Transfer turkey to a platter and tent with foil. Let rest in a warm place 15 to 30 minutes, then carve.
*Find applewood chunks with grilling supplies and at bbqislandinc.com. If pot won't fit in fridge, use an ice chest: Set bird in a turkey brining bag (buy in grocery stores or 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.