1 large egg
1 Tbs. Worcestershire sauce
2 tsp. Dijon mustard
Kosher salt and freshly ground black pepper
3/4 lb. 90% lean ground beef
1/4 lb. sweet Italian sausage, casings removed
1/2 cup plain dry breadcrumbs
1/2 small yellow onion, coarsely grated
2 large cloves garlic, finely grated
2 Tbs. vegetable oil
2 Tbs. unsalted butter
3 cups lower-salt beef broth
1 Tbs. cornstarch
8 oz. shiitake mushrooms, stemmed and thinly sliced
1 tsp. chopped fresh rosemary
2 Tbs. chopped fresh parsley
In a large bowl, whisk the egg with 2 tsp. of the Worcestershire sauce, 1 tsp. of the mustard, 1/2 tsp. salt, and 1/4 tsp. pepper. Add the beef, sausage, breadcrumbs, onion, and garlic and mix until combined. Shape into 4 oval patties about 1/2 inch thick.
Heat the oil and 1 Tbs. of the butter in a 12-inch skillet over medium-high heat until the butter stops foaming. Add the patties and cook, flipping once, until well browned and cooked through (160°F), about 10 minutes total. Transfer to a plate, and cover to keep warm. Wipe out the skillet.
In a small bowl, stir 2 Tbs. of the beef broth with the cornstarch.
Melt the remaining 1 Tbs. butter in the skillet over medium-high heat. Add the mushrooms, rosemary, and 1/2 tsp. salt and cook, stirring occasionally, until the mushrooms are browned, about 5 minutes. Add the remaining beef broth, 1 tsp. Worcestershire, and 1 tsp. mustard. Stir the cornstarch mixture again and then whisk it into the skillet. Cook, whisking occasionally, until the mixture is slightly thickened, about 5 minutes. Season to taste with salt and pepper. Return the patties to the skillet to reheat. Serve sprinkled with the parsley.