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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.
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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.
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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.
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Variations:: Chicken Livers with Caramelized Onions and Sherry Use 1/2 cup of dry sherry instead of the Madeira.
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Chicken Livers with Caramelized Onions and Port: Use 1/2 cup of port instead of the Madeira.
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Wine Recommendation: The rich and luscious Madeira sauce is ideal with fruity, spicy grenache-based wines. A Gigondas or a Cotes-du-Rhone from the Rhone Valley in France or a bottle of grenache from California would be appropriate.