Make a basic brine: In a 6-qt. or larger pot, combine the 1 cup kosher salt, 1/4 cup sugar, and 2 qt. cool water. Put the pot over high heat and stir occasionally until the salt and sugar dissolve. Remove from the heat and let cool. Stir in another 2 qt. water and chill in the refrigerator.
Soak the turkey in the brine: Remove the neck, giblets, and tail (if present) from the turkey; reserve them for making the turkey broth. Discard the liver. Rinse the turkey well. Double up two turkey-size oven bags and then roll down the edges of the bags a bit to help them stay open. Put the bags in a heavy-duty roasting pan and put the turkey, breast side down, in the inner bag. Pour the brine over the turkey (have someone hold the bags open for you, if possible). Gather the inner bag tightly around the turkey so the brine is forced to cover most of the turkey and secure the bag with a twist tie. Secure the outer bag with a twist tie. Refrigerate the turkey (in the roasting pan, to catch any leaks) for 12 to 18 hours.
Make the sage butter: In a medium bowl, stir all the ingredients until well combined. Refrigerate if making ahead.
Roast the turkey: Heat the oven to 350°F and position a rack in the lower third of the oven. Remove the turkey from the brine, rinse it very well, and pat it dry with paper towels. Discard the brine and oven bags. With your hands, gently loosen the skin from the turkey breast and legs, being careful not to tear the skin. Use one hand to distribute the sage butter under the skin and use your other hand outside the skin to massage and smooth the butter as evenly as possible over the turkey breast and as much of the legs as you can get to. Tuck the wings behind the turkey to secure the neck skin and loosely tie the legs together. Rub the turkey all over with a light coating of olive oil, and sprinkle lightly with salt (to help crisp the skin). Put the turkey, breast side up, on a roasting rack in a heavy-duty flameproof roasting pan. Put the pan in the oven, with the legs pointing to the back of the oven, if possible.
After the turkey has been roasting for 1 hour, begin rotating the roasting pan (for even browning) and basting the turkey with pan drippings every 30 minutes or so. If there aren't enough drippings to baste with at first, use a little olive oil until there are drippings. The turkey is done when an instant-read thermometer inserted in the thickest part of the thigh registers 170°F and the juices run clear when you remove the thermometer. Check the temperature in both thighs; sometimes one thigh will be done before the other. The total roasting time will be about 3 to 3 1/2 hours. Transfer the turkey to a carving board, tent it with foil, and let it rest while you make your mushroom gravy.