Whisk together the eggs, 1/2 cup cold water, the oil, and the salt. Stir in the matzoh meal slowly to avoid creating lumps. Stir until smooth and thick. Cover with plastic wrap; refrigerate until cold -- at least 2 hours. Makes 6 dozen.
In a large pot bring 4 quarts water and the salt to a boil. Place the celery, parsnip, onion, dill, and parsley sprigs in a large square of cheesecloth and tie with kitchen string. When the water comes to a boil, add the chicken, carrots, and vegetable bundle. Reduce heat to medium-low and simmer until the chicken is cooked through and the carrots are soft, about 1 1/2 hours. Remove the chicken to a bowl and reserve for another use; remove and discard the vegetable bundle.
Form the matzoh mixture into 3/4-inch balls (keep a small bowl of water nearby to prevent hands from getting sticky).
Bring broth and carrots to a simmer. Drop in the matzoh balls and simmer until they are cooked through, 10 to 12 minutes.