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Place eggs into a 2-quart saucepan. Run enough cold water into the pot to cover eggs by 2 inches. (Don't run water directly over eggs as the pressure from the tap can cause the eggs to crack.) Place pan, uncovered, over medium heat. As soon as the water begins to bubble, sprinkle in a generous teaspoon of salt. (The salt cushions the eggs from each other as the water begins to move more aggressively.) Bring water to a full boil, cover pan and remove from heat. Let pan sit undisturbed for 15 minutes. Drain and immediately run eggs under cold water until just cool enough to handle. One at a time, tap each egg with the back of a spoon all over to crack the shell. Gently roll the egg between your hands to loosen the shell. Peel off shells and discard. (The yolks of hard-cooked eggs should be bright, bright yellow (almost orange); any greenish film or pungent odor, indicates an overcooked egg.)
Melt chicken fat in a 12-inch skillet over medium-low heat. Add onions, stir to coat with the fat and cook until softened and fragrant, 5 to 10 minutes. Stir in the minced garlic and, if desired, add a pinch of sugar. Cook the onions and garlic, stirring occasionally, until the onions are golden brown and caramelized, about 25 minutes (I use really large onions, so this takes a while.).
When onions are golden, push them to the sides of the skillet and in the center melt the 2 tablespoons butter or add more fat. When hot and bubbling, add chicken livers in a single layer and sear until very golden on one side, about 5 minutes over high heat. Turn the livers over and sear briefly on the second side then cover the pan and reduce the heat to very low. Cook until the livers are just cooked through but retain some pink in the center, about 5 minutes. Uncover skillet, raise heat to high and fold the onions and livers together, scraping up any caramelized bits off the bottom of the pan. Cook about 2 more minutes over high heat. Season the mixture with salt and a generous amount of freshly ground black pepper.
Pour the contents of the skillet into a food processor fitted with the steel blade. Add hard-cooked eggs and 1 teaspoon salt. Process mixture using on-off pulses until finely chopped but still somewhat textural. Taste for seasoning, adding more salt and fresh pepper until well seasoned. Alternatively, mince cooked livers with onions on a large chopping board using a sharp chef's knife and mash the cooked eggs separately; combine livers, onions, eggs and seasoning.
Turn mixture into a crock or pretty bowl and cover with paper toweling until cool enough to refrigerate. Then cover with plastic wrap or aluminum foil and refrigerate at least 4 hours, preferably overnight, so the flavors can meld.
An hour before serving, remove chopped liver from refrigerator so that it's only slightly chilled. Sprinkle the top lightly with salt and pepper and either place some minced onion in a 1-inch border around the edge of the bowl or border the chopped liver with caramelized onions. If you’ve got them, you can also sprinkle the top with crisp Gribenes.