Preheat the oven to 375 degrees F.
Season the cavity of the chicken with salt and pepper. Tie the legs together with kitchen twine and fold the wing tips under the back. Place the onion, celery, neck, and heart in the center of a roasting pan just large enough to accommodate the chicken. (Discard the liver or reserve for other recipes.) Brush the entire chicken with the butter; season with salt and pepper. 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 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 gravy.
Pour the fat and juices in the pan to a degreasing cup or small bowl. Reserve 1 1/2 tablespoons of the fat, discard the rest, and add the roasting juices to the broth. Place the pan directly on the burner over medium-high heat and add the reserved fat. Scrape up any browned bits on the bottom of the pan with a wooden spoon. Stir in the flour and cook for 2 minutes, stirring constantly. Whisk in the broth and continue to stir. Bring to a boil and cook until thickened, about 3 minutes. Strain the sauce into a gravy boat and season with salt and pepper to taste. Discard the vegetables and giblets.
Carve the chicken and serve with gravy.
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