Combine salt, pepper, and flour together. Set aside. Combine egg and milk. Dredge hamburger patties in flour mixture then coat in milk mixture. Dredge in flour mixture again. Heat shortening in medium size skillet over medium high heat. Fry 2-3 minutes on each side until golden brown. Remove and keep warm.
Melt butter in skillet drippings. Add salt, pepper, and flour. Cook slowly on medium heat, stirring until bubbly and smooth. Add milk to the flour mixture in the skillet, stir well. Cook gravy on low heat, stirring constantly, until thickness desire is reached. Add a scoop full of gravy on top of warm steaks