1. Preheat the oven to 350F. Remove the giblets from the chicken cavity and set aside. Rinse the bird inside and out; pat dry with paper towels. Trim any excess fat. Sprinkle 1/4 teaspoon salt inside the cavity. Rub the chicken with the lemon halves, then place them in the cavity with the onion halves.
2. Place the chicken on a rack in a flameproof roasting pan and truss, if desired (see How to Truss a Chicken). Brush with the melted butter, then sprinkle with 1/2 teaspoon salt and 1/4 teaspoon pepper. Loosely cover the chicken with aluminum foil.
3. Roast the bird for 20 minutes per pound, removing the foil after 45 minutes and basting often, until an instant-read meat thermometer inserted into the thickest part of the thigh registers 180F. Transfer the chicken to a platter or carving board, cover again with foil and let rest for 15 to 30 minutes before carving. Pour the pan drippings from the pan into a measuring cup; set aside the pan and the measuring cup.
4. While the chicken is roasting, bring the water, broth, parsley and giblets (except the liver) to a simmer in a medium saucepan. Simmer 15 minutes. Add the liver; simmer 15 minutes more. Remove from the heat. Discard the neck and parsley. Remove the remaining giblets and chop; set aside. In a small bowl, whisk the flour with some of the hot broth; whisk the paste back into the saucepan.
5. Skim and discard the fat from the pan drippings in the measuring cup; return the drippings to the roasting pan. Place the roasting pan over medium heat and stir in the broth mixture. Cook, stirring and scraping the bottom of the pan to loosen browned bits, until the gravy has thickened, 5 minutes. Stir in the chopped giblets, 1/2 teaspoon salt, 1/4 teaspoon pepper and any collected juices from the platter. Transfer the gravy to a gravy boat. Carve the chicken and serve with the piping hot gravy.
