In a large bowl, combine the egg, bread crumbs, 2 tablespoons soup mix and pepper. Crumble beef over mixture and mix well. Shape into four patties.
In a large cast-iron or other heavy skillet, cook patties over medium heat until a thermometer reads 160°, and juices run clear, 4-5 minutes on each side. Set aside and keep warm.
Combine the flour, water, Worcestershire sauce and remaining soup mix until blended; stir into skillet. Add mushrooms. Bring to a boil; cook and stir until thickened, about 5 minutes. Serve sauce with patties and mashed potatoes.