How to Make It
In a small saucepan, cover the eggs with cold water and bring to a boil; cook over moderate heat for 10 minutes. Drain the eggs and immediately fill the pan with cold water. Add ice and let the eggs stand until chilled. Drain the eggs, peel and coarsely chop.
In a very large skillet, melt the butter in 1/4 cup of the chicken fat. Add the onion and cook over moderately low heat, stirring occasionally, until softened and just starting to brown, about 10 minutes. Season the livers with salt and pepper and add them to the skillet. Cook over moderately high heat, turning occasionally, until barely pink inside, about 8 minutes.
Scrape the mixture into the bowl of a food processor and let cool slightly. Add the chopped eggs and pulse until the livers are finely chopped but not completely smooth. Add the parsley and the remaining 1/4 cup of chicken fat and pulse to combine. Season with salt and pepper.
Transfer the chicken liver to a bowl. Press plastic wrap onto the surface and refrigerate until chilled, about 45 minutes. Serve with the matzo.
The chopped liver can be refrigerated overnight.
Rendered chicken fat (schmaltz) is available in the refrigerated section of most supermarkets.