Place the eggs into a saucepan in a single layer and fill with water to cover the eggs by 1 inch. Cover the saucepan and bring the water to a boil over high heat. Once the water is boiling, remove from the heat and let the eggs stand in the hot water for 15 minutes. Pour out the hot water, then cool the eggs under cold running water in the sink. Peel and chop once cold.
Meanwhile, heat the canola oil in a skillet over medium heat. Stir in the onion; cook and stir until the onion has softened and turned translucent, about 5 minutes. Reduce heat to medium-low, and continue cooking and stirring until the onion is very tender and golden brown, 5 to 10 minutes more. Place onion on a plate and set aside. Stir chicken livers into the same skillet. Cook until no longer pink in the center and the juices run clear. Place chicken livers on the same plate as the onions and allow to cool completely.
Place chicken livers and onion in a food processor. Process until desired texture is achieved. Place liver mixture into a bowl and season with salt and sugar. Stir in the chopped eggs. Chill before serving.