Drain the livers and saute them in 2 batches in 2 tablespoons of the chicken fat over medium-high heat, turning once, for about 5 minutes, or until just barely pink inside. Don't overcook the livers or they will be dry. Transfer them to a large bowl.
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 the sides, for about 15 seconds. Pour into the bowl with the livers.
Add the eggs, parsley, thyme, salt, black pepper, cayenne, and the remaining chicken fat to the bowl. Toss quickly to combine. Transfer half the mixture to the bowl of a food processor fitted with a steel blade. Pulse 6 to 8 times, until coarsely chopped. Repeat with the remaining mixture. Season, to taste, and chill. Serve on crackers or matzo.
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