Place the livers in a colander set in the sink to drain any excess liquid.
Heat a large (10- to 12-inch) sauté pan over medium-high heat for about 2 minutes, or until very hot. Add 2 Tbs. of the chicken fat. Once it has almost melted, add the onions and sauté until they start to color and soften. Lower the heat to medium and continue to sauté until the onions are lightly caramelized, but not dark brown, about 15 to 20 minutes. Remove the onions from the pan (try to leave as much fat in the pan as possible) and set aside.
Place the empty pan over high heat until it is very hot, about 2 minutes. Add the remaining 1 Tbs. of chicken fat. When it's hot, add the chicken livers. (It's important the fat and the pan are searing hot so the livers sauté rather than steam.) Season with the salt and pepper. Sauté for 4 or 5 minutes, or until the livers are cooked through. They should be brown on the outside and pink in the center. Do not overcook or the chopped liver will be dry.
Transfer the livers to a bowl and allow to cool slightly. Using a food processor, meat grinder, or the grinder attachment of an electric mixer, grind or process half the livers with half the hard-cooked eggs and half the sautéed onions. If using a food processor, pulse the mixture, stopping to scrape the sides of the bowl, until it is a coarse purée. (It should be smooth, but not the consistency of baby food.) Scrape the puréed liver mixture into a bowl.
Repeat with the remaining livers, sautéed onions, and hard-cooked eggs. Add to the mixing bowl with the first batch and taste for seasoning. Adjust with additional salt and pepper if desired.
Spoon the chopped liver into a serving bowl or individual ramekins or press it into a greased mold. Cover tightly with plastic wrap or a lid (so the surface doesn't get dried out), and refrigerate for up to 2 days. Or freeze in small portions and thaw overnight in the refrigerator. Serve with crackers or thin slices of toasted challah or brioche bread.