Prepare the Hamburger Steaks : In a large bowl, combine ground beef, breadcrumbs, egg, milk, salt, and black pepper. Mix until well incorporated. Divide the mixture into 4-6 equal portions and shape into patties. Heat a large skillet over medium-high heat and lightly oil the surface. Cook the patties for 4-5 minutes on each side or until browned. Remove from the skillet and set aside.

Caramelize the Onions : In the same skillet, add the sliced onions. Cook over medium heat, stirring occasionally, until soft and golden brown, about 8-10 minutes. Remove the onions from the skillet and set aside.

Make the Gravy : In the same skillet, reduce heat to medium and add the flour. Stir constantly for 1-2 minutes to form a roux. Gradually whisk in the beef broth to avoid lumps. Add Worcestershire sauce and dried thyme, stirring to combine. Return the caramelized onions to the skillet and let the gravy simmer for 5 minutes, thickening slightly.

Combine and Serve : Place the hamburger steaks back into the skillet, ensuring they are submerged in the gravy. Simmer for 10-15 minutes, allowing the flavors to meld and the steaks to cook through. Serve the hamburger steaks hot, topped with onion gravy, alongside mashed potatoes or rice for a complete meal.