In a bowl, mix the ground beef, breadcrumbs, finely chopped onion, egg, Worcestershire sauce, and a pinch of salt and pepper until just combined. Avoid overmixing; it keeps patties tender.

Shape the mixture into four oval patties, about 3/4-inch thick.

Heat 2 tablespoons olive oil in a large skillet over medium-high heat. When hot, add the patties and brown 3–4 minutes per side until a crust forms. Remove patties and set aside.

Reduce heat to medium. Add sliced mushrooms to the same pan and sauté 4–5 minutes until they release their moisture and start to brown.

Sprinkle 1 tablespoon flour over the mushrooms and stir for 1 minute to cook the raw flour taste.

Gradually add 2 cups beef broth while stirring, scraping up browned bits from the pan. Stir in 1 tablespoon soy sauce and 1 tablespoon ketchup. Bring to a simmer and let the gravy thicken slightly (2–3 minutes).

Return the patties to the pan, spoon some gravy over them, cover loosely, and simmer 8–10 minutes more until internal temperature reaches 160°F (71°C).

Taste the gravy and adjust seasoning with salt and pepper. Serve hot.