For the stuffing: Bring 5 cups water to a boil in a medium pot over high heat. Add 2 tsp. salt, then slowly add rice (do not stir). Reduce heat to medium, partially cover pot, and cook, shaking pot over heat occasionally, until rice is tender, 35–45 minutes. Drain off any excess liquid and transfer rice to a large bowl.
Melt 4 tbsp. of the butter in a large skillet over medium-high heat, fry livers until lightly browned, about 8 minutes, then add to rice. Wipe skillet clean. Melt remaining butter in skillet over medium-high heat, add onions and mushrooms, and sauté until onions are golden, 10–15 minutes. Add flour and cook, stirring, for 2 minutes. Add stock and cook, stirring, until thickened, 4–6 minutes, then add to rice. Whisk egg yolks and sherry together in a small bowl, add to rice, and season to taste with salt and pepper. Mix well and set aside.
For the turkey: Move oven rack to lower third of oven and preheat oven to 300°. Spoon stuffing into turkey cavity (put any excess stuffing into a small baking dish, cover, and put into oven with turkey 1 1⁄2 hours before turkey is done); then sew cavity shut using a trussing needle threaded with kitchen twine. Transfer turkey, breast side up, to a roasting rack set in a large heavy roasting pan. Moisten a large piece of cheesecloth in butter, then drape over turkey. Roast turkey, basting occasionally, until skin is golden brown and internal temperature of stuffing reaches 160°, 4–5 hours. Remove cheesecloth, transfer turkey to a platter, loosely cover with foil, and set aside to let rest for 15 minutes. Save pan drippings to make Pan Gravy, if you like. Untruss turkey, transfer stuffing to a serving dish, and carve turkey.