Place your skillet on the stovetop over medium heat and add half of the schmaltz to the pan. Once the schmaltz sizzles, add half of the chicken livers.
Add the livers to the skillet and season generously with salt and pepper. Cook the livers for approximately 3 minutes on each side until brown on the exterior but with some pink still visible in the interior. DO NOT overcook the chicken livers.
When the chicken livers are cooked, transfer them to a plate and repeat the same process with the remaining schmaltz and chicken livers.
When all the chicken livers are cooked and transferred to a plate, add the sliced onions to the skillet and cook until they are a golden brown. You should not need to add any additional fat to the skillet. This will take approximately 15 minutes.
Once the onions are cooked, transfer them to the plate with the livers and allow them to cool for a minute.
Add the livers, the onions, and 3 of the hard-boiled eggs to a food processor. Pulse the food processor in 15-second increments until the chopped chicken livers reach a consistency that you like. (See video.) After the first 15-second increment, taste the chopped livers and add additional salt or pepper if necessary.
If you do not want to use a food processor, you can chop the livers, onions, and eggs by hand with a knife. This will create more of a rough chop mixture than a smooth mixture. Alternatively, if you have a meat grinder, you can run the entire mixture through the grinder to make the chopped chicken livers.
To serve, place the chopped chicken livers in a bowl. Chop the remaining egg and use it to top the chopped chicken livers.
Chopped chicken livers will stay fresh, tightly covered and refrigerated for approximately 5 days. DO NOT freeze chopped chicken livers, as it will alter the consistency.