Make brine: Boil 1 qt. water with salt in a pot big enough to hold turkey, stirring until salt is dissolved. Add 2 qts. cold water and let cool to room temperature. Meanwhile, remove leg truss from turkey and discard. Remove neck, tail, and giblets and save for broth if you like. Pull off and discard lumps of fat. Rinse bird inside and out. Lower turkey into brine. If breast isn't submerged, make more brine, cool, and add. Chill, covered, at least 12 hours.
Make butter: In a food processor, whirl butter, sage, and garlic until smooth.
Preheat oven to 350°. Lift turkey from brine, rinse, and pat dry. Set turkey on a V-shaped rack in a 12- by 17-in. roasting pan (big enough so turkey fits inside rim).
Slide your fingers between skin and flesh of bird, turning bird to make pockets of space on the breast, back, and leg areas. Still using your fingers, slide about 1 tbsp. sage butter at a time under skin in all the pockets. Place bird, breast up, on rack.
Roast turkey, basting occasionally with pan drippings, until a meat thermometer inserted straight down through thickest part of breast to the bone registers 160°, 2 to 3 hours. Transfer turkey to a platter and tent with foil. Let rest in a warm place 15 to 30 minutes, then carve.
Meanwhile, make gravy: Pour drippings into a clear measuring cup, using a flat spatula to scrape off browned bits stuck to pan. Skim fat from drippings, reserve 1/4 cup of fat, and discard the rest. Measure drippings; add hot water if needed to make 1 cup total.
In a large frying pan, heat reserved fat over medium heat. Add flour and cook, whisking constantly, until golden brown, about 5 minutes. Add drippings, whisking into a smooth paste. Whisk in about 1/2 cup broth at a time, letting mixture come to a boil between additions. Stir in sage.
Make ahead: Chill sage garlic butter, covered, up to 4 days; bring to room temperature when ready to use.
Note: Nutritional analysis is per serving.