Rinse chicken livers, and pat dry with paper towels; sprinkle livers with 1/4 teaspoon salt and 1/4 teaspoon pepper. Heat 1 teaspoon oil in a medium nonstick skillet coated with cooking spray over medium-high heat. Add livers, and cook 4 minutes on each side or until well browned. Remove livers from pan; place in a food processor.
Wipe drippings from pan with a paper towel. Add 1 teaspoon oil to pan; heat over medium heat. Add onion and garlic; cook 5 minutes, stirring occasionally.
Add egg whites and 1/2 cup stock to food processor with livers. Process until smooth (about 1 minute); transfer to a large mixing bowl. Add onion mixture, 1/2 teaspoon salt, 1 teaspoon pepper, matzo meal, and nutmeg; stir until well blended. Cover and let stand 10 minutes. Shape mixture into 24 balls (about 1 tablespoon each).
Bring 8 cups water to a simmer in a large Dutch oven. Add matzo balls; cook 10 minutes. Remove matzo balls with a slotted spoon; keep warm.
Bring 8 cups stock to a simmer over medium heat. Add chicken and 1/2 teaspoon salt; cook 3 minutes or until thoroughly heated. Ladle 1 1/3 cups soup into each of 6 bowls; top each serving with 4 matzo balls.