Make the broth: Put the quartered hen, necks or feet (if using), onions, celery, parsley roots, carrots, turnips, parsnips, peppercorns, star anise, and salt in a very large (at least 12-quart) stockpot. Add the water and bring to a boil over high heat.

Reduce the heat to a simmer, partially cover the pot, and cook for 2 hours, skimming off any scum that rises to the surface with a large spoon.

Add the dill sprigs and simmer for another 45 minutes. Turn off the heat and allow the broth to cool. Ladle the cooled broth through a fine-mesh sieve into storage containers. Refrigerate overnight.

The next day, remove any fat that congealed on the top of the broth. Refrigerate the broth for up to 2 days, or freeze for up to 3 months.

Make the matzah balls: Combine the eggs, salt, schmaltz and the ¼ cup broth in a medium bowl and whisk until blended.

Using a fork, stir in the matzah meal, mixing until smooth. Cover and chill in the refrigerator until the matzah meal has absorbed the liquid and the mixture feels almost firm, about 30 minutes.

In a large pot, bring 2 quarts of the chicken broth to a low simmer. In a separate pot, bring about 5 quarts of water to a boil with 2 tablespoons of kosher salt.

Remove the matzah ball mixture from the fridge. Wet your hands with cold water, scoop up a heaping tablespoonful of the mixture, and shape it into a ball by gently rolling it between your palms. (Rolling too tightly will prevent it from achieving its puffiest consistency.) Drop the matzah ball into the boiling water and repeat with the remaining mixture, cleaning and wetting your hands as necessary to keep the matzah balls from sticking to them. Be careful not to crowd the pot, or the matzah balls will not cook through fully; cook in batches or in two pots, if necessary.

Bring the water back to a full boil, then cover tightly, adjust the heat to a simmer, and cook until the matzah balls have risen from the bottom of the pot and expanded to about twice their size, 35 to 40 minutes. Occasionally a matzah ball might stick to the bottom of the pot; if so, just give it a nudge with a spoon to release it.

While the matzah balls cook, make the vegetable garnish: Add the sliced carrots and diced turnips to the pot of broth and simmer for 30 minutes. Add the sliced celery and continue cooking until all of the vegetables are tender, about 15 more minutes. Add the chopped dill and taste the soup for seasoning, adding more salt if needed. Keep warm until ready to serve.

Using a slotted spoon, remove the matzah balls from the boiling water and place them directly into the simmering soup. (If your matzah balls are done before your soup is ready, let them sit on a tray, then reheat them in the soup.)

To serve, transfer the matzah balls to bowls and ladle the broth and vegetables over the top. Start with one or two matzah balls per person, inviting guests to come back for seconds of soup and/or matzah balls.