Heat chicken fat in a large skillet over medium heat. Add onions; cook, stirring often, until translucent, 8 to 10 minutes. Reduce heat to medium-low. Cook until caramelized, about 25 minutes more. Add livers. Sprinkle with 1 teaspoon salt; season with pepper. Cover; cook, stirring often, until livers are cooked through, about 12 minutes. Transfer to a bowl.
Place skillet over medium heat. Add Marsala and thyme. Cook, stirring, until reduced by half, about 2 minutes. Let cool completely. Pulse pan juices, onions, livers, and eggs in a food processor until coarsely chopped. Sprinkle with remaining teaspoon salt, and season with pepper. Serve with crackers.