Heat schmaltz and butter in a skillet over medium heat. When hot, add diced onion. Cook until softened, about 6-8 minutes.
Raise the heat to medium-high, then add cleaned chicken livers. Season lightly with Kosher salt and black pepper, then cook, stirring occasionally, until they're just barely pink inside.
Transfer to the bowl of a food processor fitted with blade attachment. Allow to cool. Add chopped hard boiled eggs, then pulse until the mixture is coarsely chopped, but not smooth. Stir in most of the parsley, then season.
Transfer to an airtight container. Press a piece of plastic wrap firmly on top of the surface of the chicken liver. Refrigerate. Serve with additional parsley, chopped hard boiled eggs, or chopped onion with crackers, matzo, or bread.