To clean the livers, first rinse them under cold running water. With a sharp paring knife, separate the two lobes of each liver. Cut away any visible fat, membranes, or green patches and drain the livers in a strainer.

Put the eggs in a small saucepan, cover with cold water by about 1 inch, and bring to a boil. Cook for 9 minutes from the moment the water boils, then remove the pan from the heat and run cold water over the eggs for a few minutes. Let the eggs cool completely, then peel and chop them; cover and set aside.

Meanwhile, heat the schmaltz in a large, 10- or 12-inch heavy skillet over medium-high heat. Add the chopped onions and sauté, stirring frequently, until soft and translucent, 7 to 8 minutes; don't let them brown. Reduce the heat to medium, add the livers, and sauté gently until they are cooked through and any liquid they release has evaporated, 15 to 20 minutes; they should still be rosy in the center. Season with the salt and pepper, remove from the heat, and let cool.

If using a wooden chopping bowl, transfer the cooked livers and onions to it. Add the mustard and horseradish and, using a mezzaluna (metal chopping blade), chop the cooked livers into a small dice. (Some of the livers will break up to mush, others will hold their shape. The goal is to obtain a spreadable combination of small pieces and paste.) If not using a chopping bowl, transfer the liver mixture to a cutting board and chop using a very sharp knife, then transfer to a medium bowl and fold in the mustard and horseradish.

Add the chopped eggs to the liver and mix gently but well, then taste and adjust the seasoning with more salt and pepper if necessary. If you like, add more schmaltz to smooth out the flavor and texture of the liver. The chopped liver can be made up to two days in advance to this point and and refrigerated, covered. Before serving, remove the liver from the refrigerator and let sit at room temperature for a few minutes, taste, and adjust the seasoning if necessary, as it may have changed after chilling.