Trim any fat or membrane from giblets. The liver should not have the gallbladder, a small green sac, attached. If it is attached, trim off carefully, removing part of liver if necessary. Do not pierce sac; the liquid it contains is very bitter. Rinse giblets and neck; pat dry.
In a medium saucepan, melt 3 tablespoons butter over medium heat. Add chopped onions, celery stalk and leaves, and leeks. Cook, stirring occasionally, until onions are translucent, about 10 minutes. Season with salt and pepper; cook another 5 minutes. Add 4 cups water, bay leaf, gizzard, heart, and neck (do not add liver; it needs to be cooked separately or it makes the stock bitter). Bring to a boil, then reduce to simmer. Cook for 45 minutes, or until gizzard is tender when pierced with the tip of a knife.
Meanwhile, chop the liver finely. Melt remaining tablespoon of butter in a small skillet over medium-low heat. Add liver and cook, stirring constantly, 4 to 6 minutes, until liver no longer releases any blood and is fully cooked. Set aside.
After 45 minutes of simmering, the liquid from the heart and gizzard stock should reduce to about 2 1/2 cups. If it has not, increase heat and cook another 10 to 15 minutes.
Strain broth. Chop gizzard and heart very fine and add to strained broth, along with chopped liver. Pick meat off neck and add to broth. Set aside until needed for gravy.