In a large bowl, whisk the egg with 2 tsp Worcestershire sauce, 1 tsp mustard, 1/2 tsp salt and 1/4 tsp pepper. Add the beef, sausage, breadcrumbs, onion and garlic and mix until well combined. Shape into 4 oval patties about 1/2 inch thick.
Heat the oil and 1 tbsp of the butter in a large skillet over medium-high heat until the butter stops foaming. Add the patties and cook, flipping once, until well browned and cooked through (about 10 minutes). Transfer to a plate and cover to keep warm. Wipe out the skillet.
In a small bowl, stir 2 tbsp of the beef broth with the cornstarch.
Melt the remaining 1 tbsp butter in the skillet over medium-high heat. Add the mushrooms and 1/2 tsp salt and cook, stirring occasionally, until the mushrooms are browned (5 minutes). Add the remaining beef broth, 1 tsp Worcestershire, and 2 tsp mustard. Stir the cornstarch mixture again and then whisk it into the skillet. Cook, whisking occasionally, until the mixture is slightly thickened (5-7 minutes). Season to taste with salt and pepper. Return the patties to the skillet to reheat.