Place the chicken in a 12- to 16-quart stockpot and add water to barely cover. Bring just to the boiling point. Then reduce the heat to a simmer and skim off the foam that rises to the top. Add all the remaining ingredients (except the optional chopped dill) and only enough water to come within about two thirds of the height of the vegetables in the pot. (Most recipes will tell you to add water to cover. Do not do this! You want elixir of the gods or weak tea? As the soup cooks, the vegetables will sink and will be covered soon enough. Eight to 10 cups of water total is plenty for this highly flavorful brew.) Simmer, covered, until the chicken is cooked through, about 1 1/2 hours.

Remove the chicken and about half the carrots from the pot, and set them aside.

Strain the soup through a fine-mesh strainer into another pot or container, pressing on the vegetables to extract all the flavor. Scrape the underside of the strainer with a rubber spatula and add the pulp to the soup. Discard the fibrous vegetable membranes that remain in the strainer. If you’re fussy about clarity (and we’re not), you can strain it again through a fine tea strainer, but there goes some of the flavor. Cover the soup and refrigerate overnight.

When you are ready to serve the soup, scoop the congealed fat off the surface and discard it. Reheat, adding more dill if desired (and we do). Slice the reserved carrots and add them to the soup. Serve the soup with matzoh balls and lukshen (thin noodles) or on Passover with mandlen (soup nuts).