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Position an oven rack in the lowest position (with room above to fit the turkey) and preheat the oven to 450 degrees F.
Remove any turkey parts from the cavity and reserve for another use if desired. Dry the turkey well with paper towels inside and out. Sprinkle the cavity with 1 teaspoon salt and 1/2 teaspoon pepper. Stuff with the thyme, rosemary, lemon wedges and half of the onion wedges.
In a small bowl, mix together the 4 tablespoons room temperature butter with the paprika to make a smooth, spreadable mixture. Using your hands, carefully separate the skin from the flesh of the turkey from the breasts toward the legs, making sure not to tear the skin. Spread about half of the butter mixture evenly between the skin and flesh on one side. Do the same on the other side using the remaining butter mixture. Any residual butter on your hands or in the bowl can be rubbed on the outside of the turkey. Sprinkle the outside of the turkey with 2 teaspoons salt, making sure to get it all over the surface.
Scatter the celery pieces, whole carrots and remaining onion wedges in the bottom of a roasting pan. Place the roasting rack in the pan, making sure it is level. Pour the broth into the pan. Transfer the turkey to the rack, tuck the wings back under the body and tie the legs together with butcher's twine.
Cut a 20-inch piece of cheesecloth and fold twice, creating 4 layers. Submerge the cloth in the melted butter, making sure it is completely saturated with it. Gently squeeze the cloth, then re-form so there are 4 layers and place on top of the turkey, covering it completely. Use a turkey baster to baste the cheesecloth with some of the remaining melted butter.
Roast for 30 minutes, then lower the oven temperature to 350 degrees F. Baste the turkey with the pan juices and more of the melted butter. Continue to roast, basting with butter and pan drippings every 30 minutes, for an additional 2 hours.
Very gently remove the cheesecloth from the turkey, taking care not to tear the skin (see Cook's Note). Baste one more time, then return the turkey to the oven with the breasts facing towards the back and roast until a thermometer inserted in the thickest part of a thigh measures 165 degrees F, about another 30 minutes. Let rest for 30 minutes before carving.