Preheat the oven to 375 degrees F.
Season the chicken cavity with salt and pepper. Tie the legs together with kitchen twine and fold the wing tips under the back. Put the carrot, onion, celery, neck, and heart, if using, in the center of a roasting pan just large enough to accommodate the chicken. (Discard the liver or reserve for another recipe.) Season the butter with salt and pepper and brush the entire chicken with it. Set the bird breast-side up on top of the vegetables.
Roast the chicken for 45 minutes. Then increase the temperature to 425 degrees F., cooking until an instant-read thermometer stuck into the chicken thigh registers 160 to 170 degrees F., about 15 minutes more. Tip the chicken so the juices run into the pan. Transfer the chicken to a cutting board and loosely tent with foil while making the jus.
To make the jus: Set a strainer over a bowl. Strain the vegetables, giblets, if using, and pan juices into the strainer. Shake the strainer to get more juices into the bowl. Set strainer aside. Skim most of the fat from the surface of the liquid with a spoon, or alternately, use a degreasing cup. Reserve the pan juices.
Place the roasting pan directly on a burner set at medium-high. Pour the chicken broth into the pan, and scrape the pan with a wooden spoon to loosen all the browned bits that cling to the pan. Add the reserved pan juices, increase the heat to high and boil the liquid until slightly thickened. Return the reserved vegetables to the sauce, if desired, or carefully strain jus through a fine sieve into a serving bowl. Taste and season with salt and pepper. Keep warm until ready to serve.
Carve the chicken and serve with the jus.
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