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Drain the livers and saute them with 2 - 3 tablespoons of the chicken fat over medium-high heat. Turn them once after about 5 minutes and cook for another 5 minutes.
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You want the livers to be just cooked through.
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Be careful not to overcook the livers or they will become dry. Transfer them to a large bowl.
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In the same pan, saute the onions in 3 tablespoons of the chicken fat over medium-high heat for about 10 minutes, or until browned. Add the Madeira and deglaze the pan, scraping up all the bits.
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Add the eggs, parsley, salt, black pepper, and cayenne to a bowl. Toss quickly to combine. Transfer each component to the bowl of a food processor working in batches. Pulse 6 to 8 times, until coarsely chopped. Repeat with the remaining mixture. Season, to taste, and chill. Serve on matzah (or crackers).
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Top with chopped eggs for appearance.