Gently mix the beef, egg, onion, cracker crumbs, sage, 1/2 teaspoon salt and a few grinds of pepper in a large bowl using your hands. Divide into 4 equal portions and shape into oval patties, about 1/2 inch thick.
Heat 1 tablespoon butter in a large skillet over medium-high heat. Brown the patties, about 4 minutes per side. Transfer to a plate.
Add 1 tablespoon butter to the skillet and cook the mushrooms, stirring occasionally, until slightly browned, about 2 minutes. Stir in the Worcestershire sauce, and salt and pepper to taste, scraping up any browned bits from the bottom of the skillet with a wooden spoon. Sprinkle in the flour and stir, then stir in the hot water and simmer until the sauce begins to thicken, about 2 minutes. Add the remaining 1 tablespoon butter, swirling to combine.
Return the patties and any juices from the plate to the skillet. Simmer until the sauce thickens and the patties are cooked through, about 2 minutes. Sprinkle with parsley and season with salt and pepper. Serve with noodles, if desired.