Melt 2 tablespoons of butter in a large skillet over medium-high heat. Add diced onions and sauté for about 3 minutes until soft.

In a mixing bowl, combine ground beef, half of the sautéed onions, eggs, Worcestershire sauce, breadcrumbs, and seasonings. Mix gently and shape into oval patties.

In the same skillet, add remaining butter. Brown patties on each side for about 2 minutes until golden crust forms. Remove and drain excess grease.

In the skillet, whisk together beef broth and flour until smooth. Bring to a simmer, then add remaining onions and Worcestershire sauce. Cook until gravy thickens.

Return browned patties to skillet, nestling them into the gravy. Simmer for 15-20 minutes, spooning gravy over patties. Serve hot.