In a medium bowl, combine the eggs, matzo meal, chicken fat or shortening, and salt. Add enough of the 1/3 cup water to make a stiff batter (mixture thickens some as it stands). Cover; chill, while broth cooks.
Rinse chicken pieces. In a large kettle, combine chicken pieces and the 2 quarts water. Cover; bring to boiling. Add carrots, onion, celery, parsley, garlic, salt, and pepper. Heat to boiling; reduce heat. Cover and simmer for 1 hour or until chicken is very tender. Reserve meat from chicken pieces for another use; discard vegetables. Strain broth and skim off any fat.
Meanwhile, with your moistened hands, form matzo batter into sixteen 1-1/4- to 1-1/2-inch balls. Carefully drop the matzo balls into boiling, lightly salted water in a large saucepan. Cover and simmer for 30 minutes.
Reheat broth. Ladle hot chicken broth into 8 warm soup bowls. With a slotted spoon, transfer 2 matzo balls per bowl. Makes 8 side-dish servings.