Put all the ingredients, except the matzo balls, in a large stockpot, cover abundantly with water, and bring to a boil. Skim to remove all the gray scum that will float to the surface, then let cook at a simmer for about 3 hours. Just keep tasting: when the broth tastes golden and chickeny, it's ready. Remove the chicken and, if you like, leave the soup to get cold to you can remove any fat that collects on the surface. That way you can accrue some schmaltz, too.
Reheat the stock, and serve it as a plain soup, or add a few carrot sticks - from about 2 carrots, say - and cook in the soup, adding some torn-up pieces of chicken to warm through at the end. I like to add freshly chopped parsley. (You can also cook the Matzo Balls in the soup as well while it's heating up again.)
Whisk the egg in a large-ish bowl, then whisk in the melted schmaltz (or whatever). Carry on whisking as you add the water or soup stock, the matzo meal and salt and pepper, and mix together into a rough paste; if it's too stiff to feel that it might be malleable later, add a little more water. Put in the refrigerator to chill for 1 hour (or leave overnight if you wish) then dip out small lumps of paste and roll them into walnut-sized balls between the palms of your hands. Cook the dumplings in boiling, salted water and simmer for about 40 minutes (you can just cook them directly in the soup, but I'll do anything to preserve its unstarchy clearness). They are cooked when they rise to the surface. Add to the soup, and ladle out generously into waiting bowls.
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