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In a large frying pan, heat 2 tablespoons of the oil over medium heat. Add onions, 1/2 teaspoon of the salt, and 1/8 teaspoon of the pepper. Cook, stirring frequently, until onions are well browned, about 35 minutes. Remove onions from the pan and transfer to 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 chicken livers with the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. Put livers in pan, in two batches if necessary, and cook for 2 minutes. Turn over and cook until browned, about 2 minutes longer; livers should still be pink inside. Remove chicken livers from the pan and place on top of onions.
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Return pan to stove, increase heat to high and add Madeira. Boil rapidly, scraping bottom of the pan to dislodge any brown bits, for 1 minute. Pour sauce over chicken livers and onions. Top with egg and parsley.
© Ben Dearnley
Notes
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.
Suggested Pairing
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.