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 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.
Prepare a gas grill for indirect low heat (250° to 300°): Set an aluminum drip pan (at least as large as turkey) under the cooking grate and on middle burner for a 3-burner grill or on one burner for a 2-burner grill. Fill drip pan halfway with hot tap water (this is the indirect heat area). With grill lid open, ignite direct heat area and turn to medium (about 350°). Drain 2 soaked wood chunks and set directly on burners (not under the drip pan). Close lid and heat 10 minutes, then adjust to maintain 250° to 300°.
Once smoke is coming from grill (it may take 20 minutes), set turkey with breast up on cooking grate over drip pan. Close lid; cook and smoke, setting 2 drained wood chunks directly on a lit burner every 30 minutes and adding water to drip pan if needed to keep it about half full, until meat thermometer registers 165° to 170°, 2 1/2 to 3 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.