Step 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.

Step 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.

Step 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.