Mix salt, brown sugar, and pepper together in a small bowl.
Rinse turkey in cold water; pat dry inside and out. Place turkey in a shallow baking pan. Rub outside with half of the salt mixture. Separate skin from the breast meat; rub remaining salt mixture between skin and breast meat. Place uncovered in the refrigerator, 8 to 12 hours.
Remove turkey from refrigerator. Let sit until room temperature, about 1 hour.
Preheat oven to 450 degrees F (230 degrees C).
Rinse turkey with cold water to remove salt mixture; place into a roasting pan. Stuff turkey cavity with celery, onion, and carrots. Separate turkey skin from the breast meat; rub softened butter onto breast meat. Pour chicken broth over the turkey.
Bake turkey in preheated oven, breast side up, for 30 minutes. Reduce heat to 325 degrees F (165 degrees C); cook, basting occasionally, until no longer pink at the bone and the juices run clear, about 2 hours. An instant-read thermometer inserted into the thickest part of the thigh should read 165 degrees F (74 degrees C). Remove turkey from the oven, cover with a doubled sheet of aluminum foil, and allow to rest in a warm area 30 minutes before slicing.