1. In a large frying pan, 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 browned, about 15 minutes. Remove the onions from the pan and put on a serving platter or individual plates.
2. In the same frying pan, 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 should still be pink inside. Remove the chicken livers from the pan and put them on top of the onions.
3. Return the pan 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.
Variations:: Chicken Livers with Caramelized Onions and Sherry Use 1/2 cup of dry sherry instead of the Madeira.
Chicken Livers with Caramelized Onions and Port: Use 1/2 cup of port instead of the Madeira.
Wine Recommendation: The rich and luscious Madeira sauce is ideal with fruity, spicy grenache-based wines. A Gigondas or a Côtes-du-Rhône from the Rhône Valley in France or a bottle of grenache from California would be appropriate.