In a large bowl, combine the ground beef, breadcrumbs, egg, minced onion, minced garlic, Worcestershire sauce, thyme, paprika, salt, and pepper. Gently mix with your hands until just combined—do not overmix. Divide the mixture into 4 equal portions and shape each into a 3/4-inch thick oval patty.
Heat the vegetable oil in a large skillet (preferably cast iron or stainless steel) over medium-high heat. Once hot, add the patties. Cook for 4-5 minutes per side, or until a deep brown crust forms and they are cooked to your desired doneness (160°F for well-done). Transfer the cooked patties to a plate and tent loosely with foil.
Reduce the heat to medium. In the same skillet with the beef drippings, melt the butter. Sprinkle the flour over the butter and cook, whisking constantly, for 2-3 minutes until the mixture (roux) turns a rich, nutty brown color.
Slowly pour in the beef broth while whisking constantly to prevent lumps. Add the Worcestershire sauce, onion powder, garlic powder, and thyme. Bring to a simmer, whisking frequently, and cook for 5-7 minutes until the gravy thickens to a coating consistency. Season with salt and pepper to taste.
Return the hamburger steaks to the skillet, nestling them into the gravy. Reduce heat to low and let them simmer gently in the gravy for 3-4 minutes to reheat and absorb flavor.
Serve each hamburger steak smothered with gravy over a generous portion of mashed potatoes. Spoon extra gravy over the top. Garnish with chopped parsley if desired.