Combine flour, 1/4 teaspoon salt, and 1/8 teaspoon pepper; mix well. Dredge livers in flour mixture; brown in 3 tablespoons butter. Cook over medium heat 5 minutes or until livers are done.
Combine eggs, water, and remaining salt and pepper; beat well. Heat a 10-inch omelet pan or skillet until hot enough to sizzle a drop of water. Add remaining 1 tablespoon butter; rotate pan to coat bottom.
Pour egg mixture quickly into pan. Shake pan vigorously (or use spatula to lift cooked portion) so that uncooked portion flows underneath. Slide pan back and forth over heat to keep mixture in motion.
Spoon livers over half of omelet when eggs are set and top is still moist and creamy. Fold unfilled side over filling. Serve omelet immediately.