Heat the butter in a frying pan over medium heat. Stir in the onion; cook and stir until the onion has softened and turned translucent, about 5 minutes. Add the liver; cook and stir until the liver is evenly browned, 5 to 7 minutes. Remove from heat and cool until just warm.
Place challah bread in a large bowl. Toss with the onion and liver mixture, squeezing lightly to distribute any liquid. Stir in the beaten eggs and season with salt.