Mix beef, breadcrumbs, egg, milk, chopped onion, Worcestershire, mustard, salt, and pepper until just combined; form into 4 oval patties.
Heat oil in a skillet over medium-high and brown patties 3–4 minutes per side; remove and set aside.
Add butter and sliced onion to skillet, cook until soft and golden, about 6 minutes.
Stir in flour and cook 1 minute, then slowly whisk in beef broth and soy sauce; simmer until slightly thickened.
Return patties to pan, cover, and simmer 8–10 minutes until cooked through and gravy coats them.