In a large skillet, heat 2 tablespoons of the oil over moderate heat. Add the onions, 1/2 teaspoon of the salt, and 1/8 teaspoon of the pepper. Cook, stirring frequently, until the onions are well cooked, about 15 minutes. Remove onions to a serving platter or individual plates.
In the same skillet, heat the remaining 1 tablespoon oil over moderately high heat. Season the chicken livers with the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. Put the livers in the pan (in two batches if necessary) and cook for 2 minutes. Turn and cook until browned, about 2 minutes longer.
The livers will still be pink inside. Remove the chicken livers from the pan and put them on top of the onions.
Return the skillet to the heat and add the Madeira. Boil rapidly, scraping the bottom of the pan to dislodge any brown bits, for 1 minute. Pour the sauce over the livers and the onions. Top with the egg and parsley.
Notes
SERVING SUGGESTION: Garlicky Stir-Fried Baby Spinach Heat 1 tablespoon olive oil in a large skillet over medium heat; add 1 clove pressed garlic and cook for 1 minute. Gradually add 1 (10-oz) package baby spinach; cook, turning with tongs, until slightly wilted; drizzle with a little balsamic vinegar and sprinkle with salt and pepper to taste.NUTRITION: 401 Calories; 18g Fat; 30g Protein; 21g Carbohydrate; 3g Dietary Fiber; 717mg Cholesterol; 500mg Sodium. Exchanges: 0 Grain (Starch); 4 Lean Meat; 2 ½ Vegetable; 2 Fat.