Remove the giblets and neck from the turkey, then rinse. In a clean bucket or other container large enough to hold the turkey, mix until the salt and 2 gallons of water dissolve.
Submerge the turkey in the solution. If the turkey is not completely covered, prepared additional brine using a ratio of 1 pound salt to 1 gallon water. Set the turkey in a very cool spot for 4 to 6 hours (perhaps inside a large cooler with ice). If it suits your schedule better, you can decrease the salt by half (in proportion to the water) and soak the turkey for 12 to 18 hours.
Position a rack at the lowest level of the oven. Preheat the oven to 325 degrees.
Remove the turkey from the brine. Thoroughly rinse inside and out, then pat the skin and the cavity dry. Place the onion, carrot, celery and thyme in the cavity.
Truss the bird. (See cook's notes)
Brush the skins all over with 4-6 tbsp melted butter. Place a V-rack or sturdy wire rack in a roasting pan and arrange the turkey breast side down on the rack. If you are using a flat rack and the turkey topples over, prop it up with balls of aluminum foil. Pour into the pan 3/4 cup of water.
Roast the turkey breast side down for 2 hours if it weighs 18 pounds or less, 2 1/2 hours if it weighs between 18 and 21 pounds, and 3 hours if it weighs more than 21 pounds. Baste the back and legs once or twice with 2 tablespoons of butter.
Remove the turkey from the oven. Protecting your hands wth paper towels, grasp the turkey at both ends and turn breast side up. Return the turkey to the oven and roast, basting once or twice with pan drippings, until an instant read thermometer plunged into the thickest part of the thigh registers 175 to 180 degrees, 30 to 90 minutes more, depending on the turkey's size. (If the turkey approaches doneness before the breast has browned, increase the oven temp to 400 degrees for the last 5 to 10 minutes of roasting.) Remove the turkey to a platter and let stand for 20 to 40 minutes.