Preheat the broiler.
Hard-Boil the eggs, cool, and peel. Cut the eggs into eighths and set aside.
In a 10- to 12-inch heavy skillet, warm 3 tablespoons of the oil over medium heat. Add the sliced onions, salt and pepper them lightly, and sauté, lifting and turning them occasionally, until softened and translucent, about 10 minutes. Cover tightly and cook the onions over the lowest heat, stirring occasionally to prevent burning, for 40 to 45 minutes, or until meltingly tender. Uncover, add the vinegar (if the balsamic vinegar is rough or strong, use 1 tablespoon; if it is soft and mild, use 2 tablespoons) and additional salt and pepper to taste, and cook over high heat, lifting and tossing, until the moisture has evaporated and the onions are colored a rich brown, about 15 minutes. Taste to adjust seasoning: the mixture should be just slightly sweet, but well salted and peppered, and the vinegar scent should be almost gone. If necessary, add the brown sugar and additional seasoning, and continue sautéing until thoroughly incorporated. Transfer about ¾ of the mixture to a food processor, add 2 tablespoons oil, and puree.
Meanwhile, prepare the liver: line the broiler rack with either heavy brown paper sprinkled with water, or foil. Pat the livers dry with paper towels and spread them out on the broiler rack. Sprinkle them lightly with salt, and broil about 4 inches from the flame until lightly browned on top, 3 to 4 minutes. Turn, sprinkle the other side with salt and continue broiling for another 3 to 4 minutes. Add the broiled livers to the cooked onions remaining in the skillet and sauté for a minute or two. Let cool slightly, then add to the onion puree in the food processor. Pulse on and off to chop coarsely. Add the eggs. Pulse. Add the chopped raw onions, and pulse on and off until desired texture is achieved. Transfer to a large bowl, taste for seasoning, and add enough oil or schmaltz (1 to 3 tablespoons or more, if needed) to make it moist and rich. Mix well and refrigerate, covered, until thoroughly chilled.
Serve cold on lettuce or radicchio or alternating green and red Belgian endive leaves. Or pack into small custard cups or cleaned tuna cans and invert onto frilly greens for a sophisticated presentation.
Accompany the liver with the suggested vegetables and breads. It is particularly good with grated black radish and endive salad served alongside, topped with chopped red radishes, or sprinkled with griebenes reserved from preparing Poultry Schmaltz.