Gently combine the ground beef, breadcrumbs, egg yolk, minced onion, Worcestershire sauce, garlic powder, salt, and pepper in a large bowl. Avoid overmixing so the patties stay tender.

Divide the mixture into four equal portions and shape each portion into an oval patty about 3/4 inch thick.

Heat the vegetable oil in a large skillet over medium-high heat. Brown the patties for approximately 3–4 minutes per side, then transfer them to a plate.

Reduce the heat to medium and add the butter, sliced onion, and mushrooms. Cook for 5–7 minutes, stirring occasionally, until the vegetables are softened and lightly browned.

Sprinkle the flour over the vegetables and stir for about 1 minute to coat them and cook off the raw flour taste.

Gradually whisk in the beef broth, scraping up the browned bits from the bottom of the skillet. Stir in the ketchup, Worcestershire sauce, and yellow mustard.

Return the patties and any accumulated juices to the skillet. Cover and simmer gently for 10–15 minutes, turning the patties once, until the gravy thickens and the beef reaches 160°F (71°C) internally.

Taste the gravy and adjust the salt and pepper. Serve the Salisbury steaks with mashed potatoes, egg noodles, or rice.