In a large bowl, combine the beef, egg, breadcrumbs, Worcestershire sauce, and ½ teaspoon each salt and pepper. Mix well.
Shape the meat mixture into 4 even balls.
Heat 2 tablespoons oil in a large skillet over medium-high heat. Add the meat balls to the skillet and use a spatula to flatten each one slightly. Cook, flipping once, until browned on each side, about 5 minutes per side. Transfer to a plate. (The meat will still be raw inside. That's ok.)
Reduce the heat to medium low and add the remaining 1 tablespoon oil to the skillet. Stir in the onion and cook, stirring occasionally, until softened, 8 to 10 minutes.
Add the mushrooms and cook until they begin to release their liquid, 5 to 6 minutes.
Sprinkle the flour over the mushrooms and cook, stirring constantly, for 1 to 2 minutes.
Slowly pour in the broth, stirring constantly. Stir in the Dijon mustard.
Bring the gravy to a simmer, then nestle the beef patties in the gravy. Simmer gently, turning the patties once, until cooked through to an internal temperature of 160°F, 15 to 20 minutes.
Season to taste with salt and pepper and serve hot.