Chicken Livers with Caramelized Onions and Madeira

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.

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.

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.