Mix egg, bread crumbs, salt, and black pepper together in a large bowl. Break ground beef into small pieces and add to the bowl; mix well with your hands until beef is evenly moistened. Shape beef mixture into four patties.

Heat a skillet over medium heat. Cook patties in hot skillet until completely browned, 2 to 3 minutes per side; remove to a platter and drain grease from skillet.

Return skillet to medium heat. Stir tomato sauce, green onions, brown sugar, Worcestershire sauce, and yellow mustard together in the skillet. Gently add patties into the sauce and bring to a boil; reduce heat to medium-low and simmer until patties are very firm, hot, and cooked through, about 10 minutes more. An instant-read thermometer inserted into the center should read 160 degrees F (70 degrees C).