In a large bowl, combine the ground beef, breadcrumbs, beaten egg, minced onion, minced garlic, Worcestershire sauce, smoked paprika, dried thyme, 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, pressing a slight indentation into the center of each to prevent doming during cooking.
Heat the vegetable oil in a large skillet (preferably cast iron or stainless steel) over medium-high heat. Once the oil is shimmering, carefully add the patties. Cook for 4-5 minutes per side, without moving them, until a deep golden-brown crust forms. Transfer the cooked patties to a plate and set aside. Do not clean the skillet.
Reduce the heat to medium. Add the butter to the same skillet with the beef drippings. Once melted, sprinkle in the flour. Whisk constantly for 2-3 minutes, scraping up any browned bits from the bottom of the pan, until the mixture (the roux) turns a rich, nutty brown color.
Slowly pour in the beef broth while whisking vigorously to prevent lumps. Add the Worcestershire sauce, onion powder, and garlic powder. Bring the gravy to a simmer, whisking frequently, until it thickens enough to coat the back of a spoon, about 5-7 minutes. Season with salt and pepper to taste.
Return the hamburger steaks and any accumulated juices to the skillet, nestling them into the gravy. Reduce heat to low, cover, and let simmer gently for 8-10 minutes to allow the flavors to meld and the steaks to finish cooking through.
Serve the hamburger steaks hot, generously ladled with the brown gravy. Garnish with chopped parsley if desired. They are traditionally served over a bed of creamy mashed potatoes with a simple vegetable on the side.