To Make the Broth

Put the chicken in a large stockpot and cover with the cold water. Bring to a bare simmer over medium heat, skimming off the foam that rises to the top.

Add the onions, 2 whole carrots, celery, parsley, 6 dill sprigs, peppercorns and 1 tbsp salt. Keep it at a gentle simmer, partly covered, for 1 1/2 hours, until the chicken is falling off the bone.

Lift the chicken out and set it aside to cool. Strain the broth through a fine sieve into a clean pot and discard the vegetables and herbs. If you want to use the chicken fat for the matzo balls, chill the broth first and lift the fat off the top, or just use oil.

Pull the meat off the bones, discarding the skin and bones, and shred it. Slice the 2 remaining carrots into coins. Add the carrots to the broth and simmer for 15 minutes, until tender. Return the shredded chicken to the pot and taste for salt.

To Make the Matzo Balls

While the broth simmers, beat the eggs and schmaltz together in a bowl. Stir in the matzo meal and salt, then the seltzer or broth, just until combined. Do not overmix. Cover and refrigerate for at least 1 hour.

Bring the 4 quarts of water and 1 tbsp salt to a boil in a large pot. With wet hands, roll the chilled batter into 12 balls about 1 1/2 inches across. For fluffier matzo balls, roll them gently without pressing. For denser ones, roll them firm.

Drop the balls into the boiling water, reduce to a simmer, cover and cook for 30 to 40 minutes, without lifting the lid, until they are cooked through and have doubled in size. Cut one in half to check: it should be the same color all the way through.

To Serve

Reheat the soup. Use a slotted spoon to move 2 matzo balls into each bowl, ladle the hot broth, carrots and chicken over them, and top with the chopped dill.