Mix ⅓ cup (packed) dark brown sugar, ⅓ cup Diamond Crystal or ¼ cup Morton kosher salt, 2 Tbsp. chili powder, 2 Tbsp. smoked paprika, 1 Tbsp. freshly ground black pepper, 1 Tbsp. garlic powder, 1 Tbsp. ground coriander, and 1 Tbsp. onion powder in a medium bowl to combine. Place one 12–14-lb. turkey, neck and giblets removed, patted dry, breast side up, on a wire rack set inside a large rimmed baking sheet. (If you don’t have this setup, place turkey on a V-shaped rack set inside a large roasting pan.) Sprinkle dry brine all over outside and inside of turkey, patting to adhere and nudging some into crevices. Chill bird, uncovered, at least 12 hours and up to 2 days.
Let turkey sit at room temperature 1 hour before cooking.
Stack two 16×12″ pieces of heavy-duty foil on a surface and mound 2 cups wood chips on top. Bring long sides of foil together and fold up tightly to seal; fold short sides under to create a pouch. (Folding the sides under will create extra layers of protection from the heat.) Repeat process with more foil and remaining wood chips to make 2 more pouches. Using a paring knife, poke about 15 holes in top of each pouch.
If using a gas grill: Prepare grill for medium-low indirect heat (set one burner to medium-high heat; leave remaining burners off). Place a wood-chip pouch directly over flame; place a drip tray on cooler side of grill. Cover grill and heat to 300° (place an oven thermometer in the center of indirect side of grill if there isn’t one built in). Lightly oil grate.
If using a charcoal grill: Fill chimney starter with charcoal; light and let burn until coals are covered with a thin layer of ash. Pour out coals and bank to cover one third of grill. Place a wood-chip pouch directly on top of coals; set a drip tray on cooler side of grill. Place grate on grill, cover, and heat to 300°, adjusting vents on top and bottom of grill as needed to control temperature (place an oven thermometer in the center of indirect side of grill if there isn’t one built in). Lightly oil grate.
If using a smoker: Prepare smoker and heat to 300° according to manufacturer’s directions. Lightly oil grate.
Meanwhile, place 1 medium red onion, quartered, inside turkey cavity. Using your fingers, gently separate skin from breast meat starting at neck. Work 4 Tbsp. unsalted butter, room temperature, under skin, spreading evenly over both breasts. Tie legs together with kitchen twine and tuck wings underneath.
Place turkey, breast side up, over indirect heat and cover grill. Grill turkey, basting with melted butter (from remaining ¾ cup (1½ sticks) unsalted butter, melted) every 30 minutes, rotating bird 180° every hour, and replacing wood-chip pouch whenever smoke dissipates, until skin is golden brown and an instant-read meat thermometer inserted into the thickest part of turkey breasts registers 145°, 3¼–3¾ hours. Keep an eye on the temperature inside the grill and aim to maintain heat at 300° as much as possible. If using a charcoal grill, light a chimney of coals and add as needed.
Rotate turkey so breast is as far from direct heat as possible and continue to grill, still covered, until skin is deeply browned and an instant-read meat thermometer inserted into the thickest part of the thighs registers an internal temperature of 165° (breast should register 155°), 30–60 minutes longer.
Transfer turkey to a cutting board and let rest at least 30 minutes and up to 1 hour before carving.