Heat a large nonstick skillet over medium-high heat. When hot, add 2 tablespoons of the chicken fat and sauté half of the chicken livers until golden around the edges and barely pink inside, about 5 minutes or so per batch. Transfer the livers to a large bowl and set aside. Repeat with the remaining chicken livers, adding an additional 2 tablespoons of fat to the skillet.

When all of the livers have been sautéed, add 2 more tablespoons of the fat to the pan and add the onions. Season lightly with salt and pepper and cook, stirring occasionally, until the onions are very soft and caramelized around the edges, about 8 minutes. Add the garlic and cook until fragrant, 1 to 2 minutes. Add the Madeira and cook, scraping to release the browned bits on the bottom of the skillet, until the Madeira is nearly completely evaporated, 2 to 3 minutes.

Remove the skillet from the heat and add the onion mixture to the bowl with the chicken livers along with the remaining chicken fat, 4 of the eggs, 2 tablespoons parsley, the thyme, and season lightly with salt and pepper. Stir to combine and then working in 2 or 3 batches, transfer to a food processor and pulse until chopped to the desired consistency.

Season to taste with salt and pepper and transfer to a serving bowl. Garnish the top with the remaining chopped egg and additional parsley. Serve either warm or at room temperature, with croustades, crackers, or matzoh.