Whisk together the eggs, onion, beef base, and pepper in a bowl. Tear the bread into large pieces, and place into the bowl. Let the bread soak for a few seconds and add the ground beef. Mix the meat into the egg-bread mixture, and form into 8 patties.

Heat a large heavy skillet over medium-high heat, and fry the patties until they are browned and the meat is no longer pink in the center, about 8 minutes per side. Set the patties aside.

Remove the excess grease from the skillet, and whisk the mushroom soup, water, and Worcestershire sauce together in the skillet until smooth. Place the patties back into the skillet, spoon sauce over them to cover, and bring to a boil over medium-high heat. Reduce the heat to a simmer, and cook until the sauce is thickened and the flavors have blended, about 20 minutes.