1. Place chicken and the 10 cups water in large saucepot. Bring to boil; reduce heat. Cover and simmer 30 minutes; skim broth. Add vegetables, garlic, peppercorns, 2 parsley sprigs, and the 4 teaspoons salt; cover and simmer 2-1/2 hours. Cool slightly and remove chicken. (Reserve chicken for another use.)
2. Strain soup; reserve a few carrot pieces and cut into thin slices for garnish. Refrigerate overnight. Skim off fat.Make Matzo Balls::
3. Combine eggs, the 1/2 cup water, and the melted fat in medium bowl. Stir in matzo meal, the 1 teaspoon salt, the pepper, and nutmeg. Cover and refrigerate 1 hour.
4. Bring 1-1/2 quarts water to boil in medium saucepan. With wet palms, roll matzo mixture into 1-inch balls and drop into boiling water. Simmer, covered, 20 minutes. Remove with slotted spoon and add to soup. Heat through. Garnish with remaining parsley sprig and reserved carrot slices, if desired. Makes 10 cups soup and 20 Matzo Balls.