In a large bowl combine ground beef, breadcrumbs, milk, egg, 1 tablespoon Worcestershire sauce, salt, and pepper. Mix gently until just combined; form into 4 even patties.
Heat vegetable oil in a large skillet over medium-high heat. Add patties and cook 4 to 5 minutes per side until browned and cooked through. Transfer patties to a plate and keep warm.
Reduce heat to medium and add butter to the same skillet. Add the sliced onion and cook, stirring occasionally, until soft and golden, about 8 to 10 minutes.
Sprinkle flour over the cooked onions and stir to coat; cook 1 minute to remove raw flour taste.
Gradually whisk in beef broth, scraping up browned bits from the pan. Add 1 teaspoon Worcestershire sauce, bring to a simmer, and cook until gravy thickens, about 3 to 5 minutes. Taste and adjust seasoning.
Return the hamburger steaks to the skillet and spoon gravy over them. Simmer 2 to 3 minutes to reheat and meld flavors. Serve hot with mashed potatoes or your preferred side.
Notes
Leftover gravy keeps well refrigerated for 3 days and freezes for up to 3 months. Reheat gently and add a splash of broth if too thick.