Combine beef, black pepper, dry mustard, ketchup, Worcestershire, soy sauce, egg, cayenne, and bread crumbs in a mixing bowl. Use a fork or your hands to stir the ingredients until very evenly distributed.

Cover with plastic wrap and refrigerate for 2 hours, or up to overnight.

While the meat is resting, melt 2 tablespoons butter in a large nonstick pan over high heat; add mushrooms, onions, and a pinch of salt. Saute, stirring occasionally, until very nicely browned, 5 to 8 minutes. Reduce heat to medium and add flour. Cook and stir for 3 minutes.

Stir in ketchup, Worcestershire sauce, and beef broth; increase heat to high. Bring to a simmer; reduce heat to medium and cook, stirring occasionally, until the gravy thickens slightly and reduces, about 10 minutes. Taste and adjust salt as needed. Transfer to a glass measuring cup, scraping the pan clean with a spatula, and reserve.

Remove meat from the refrigerator and divide into 4 equal portions. Moisten hands and form meat into oval patties, about 1/2 inch thick. Season both sides of the patties lightly with salt.

Melt 2 tablespoons butter in the previously used nonstick pan over medium-high heat. Add the patties and cook until browned, 2 to 3 minutes per side. Pour in gravy and bring to a simmer; reduce heat to medium and continue to cook until gravy is piping hot, and the meat is just cooked through and springs back lightly when touched, about 5 minutes.