Heat oven to 400° F. Stuff the turkey cavity with the sage and 4 onion quarters. Scatter the carrots and the remaining onions in a large roasting pan.
Place the turkey on top of the onions and carrots. Tie the legs together with kitchen twine and tuck the wings underneath the body. Brush the turkey with the butter and season with 1 teaspoon salt.
Roast the turkey for 45 minutes. Add the broth to the roasting pan. Continue to roast the turkey, covering it loosely with foil if it browns too quickly, until a thermometer inserted into a thigh registers 165° F, 2 to 2 1/4 hours.
Carefully tilt the turkey to empty the juices from the cavity into the pan. Transfer the turkey to a cutting board, tent loosely with foil, and let rest for at least 20 minutes before carving. Discard the onions and carrots. Reserve the pan drippings for Hard Cider Gravy.
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