Place the chicken livers on a cutting board and trim away any visible fat, connective tissue, or greenish spots. Pat very dry with paper towels so they brown instead of steam. Cut any very large pieces in half so they cook evenly.

In a large skillet, heat 2 tablespoons schmaltz (or neutral oil) over medium heat. Add the sliced onions and a pinch of salt. Cook, stirring often, for 15–20 minutes, until the onions are soft, golden, and lightly caramelized. Lower the heat if they start browning too quickly.

Stir the minced garlic into the onions and cook for about 1 minute, just until fragrant. Scrape the onion–garlic mixture into a bowl and set aside. Do not wash the pan.

In the same skillet, add the remaining 1 tablespoon schmaltz and the butter (if using). Raise the heat slightly to medium-high. Add the chicken livers in a single layer, without crowding; cook in two batches if necessary. Sear for about 3–4 minutes per side, until the outsides are browned and the centers are just barely pink (about 165°F at the thickest part). Do not overcook.

While the livers are still in the pan and warm, sprinkle with about 1 teaspoon kosher salt, the black pepper, and the sweet paprika. Toss gently to coat so the seasoning absorbs into the warm livers.

Transfer the seasoned livers to a large bowl or the bowl of a food processor and let cool for 10–15 minutes. They should be warm but not steaming hot before you add the eggs.

Add the caramelized onions and garlic to the bowl with the livers. Add the hard-boiled eggs (roughly chopped if you like). Taste a small bite of onion and liver together and adjust the salt and pepper slightly if needed at this stage.

For a rustic chopped liver, transfer everything to a cutting board and chop together with a large knife or mezzaluna until finely chopped but still slightly chunky. For a smoother pâté-style spread, pulse the mixture in a food processor 6–10 times, scraping down the sides as needed. For extra smoothness, add 1–2 tablespoons chicken stock or water and 1–2 teaspoons additional schmaltz or olive oil while processing.

Taste a small spoonful. It should be savory, gently sweet from the onions, and well-seasoned but not overly salty. Add more salt, pepper, or paprika as needed. If the flavor tastes flat, a pinch of salt and another teaspoon of schmaltz can brighten it.

Transfer the chopped liver to a serving bowl or storage container. Smooth the top and cover tightly. Chill in the refrigerator for at least 2 hours, preferably overnight, to allow the flavors to develop and the texture to firm for easy spreading.

Before serving, let the chopped liver sit at room temperature for 15–20 minutes so it’s not ice-cold. Garnish with chopped fresh parsley and/or sliced scallions or chives and a light dusting of paprika if you like. Serve with matzo, rye or pumpernickel bread, crackers, sliced cucumbers, pickles, or crisp lettuce leaves.