Put the chicken and enough water to cover by two inches (about 4 quarts) in a large pot and bring the water to a boil. Skim off the froth as it rises to the top.
Add the onions, parsnips, celery, carrots, parsley, 4 tablespoons of the dill, and the salt and pepper. Half-cover and simmer for at least an hour and up to 2 hours, adjusting the seasoning to taste.
Refrigerate for 2 to 3 hours or overnight so the liquid solidifies. When the fat rises to the top, skim it off and reserve for the matzo balls.
For the matzo balls
Using a spoon, gently mix the eggs, schmaltz, stock, matzo meal, nutmeg, ginger, and parsley, dill, or cilantro in a large bowl. Season with salt and 2 to 3 grinds of pepper. Cover and refrigerate until chilled, at least an hour or overnight.
To shape and cook the matzo balls, bring a wide, deep pot of lightly salted water to a boil. With wet hands, take some of the mix and mold it into the size and shape of a golf ball. Gently drop it into the boiling water, repeating until all the mix is used.
Cover the pan, reducing heat to a lively simmer. Cook for about 20 minutes for al dente matzo balls, or closer to 45 for lighter matzo balls. To test their readiness, remove one with a slotted spoon and cut in half. The matzo ball should be the same color and texture throughout.
Strain the soup. Set aside the chicken for chicken salad and discard the vegetables. Just before serving, reheat the soup. Spoon a matzo ball into each bowl, pour soup over each matzo ball, and sprinkle with the remaining snipped dill.