Mix the ground beef, egg, breadcrumbs, minced onion, Worcestershire sauce, garlic powder, kosher salt, and black pepper just until combined. Avoid overmixing because compacted beef can produce tough patties.

Divide the mixture into four portions and shape each portion into an oval patty about 3/4 inch thick. Handle the mixture lightly and keep the patties evenly sized so they cook at the same rate.

Heat a large skillet over medium-high heat. Brown the patties for approximately 3–4 minutes per side, then transfer them to a plate; they do not need to be fully cooked yet.

Reduce the heat to medium and add the butter or vegetable oil. Sauté the sliced mushrooms for 4–5 minutes, stirring occasionally, until browned and their moisture has cooked off.

Sprinkle the flour over the mushrooms and stir for about 1 minute. Gradually whisk in the beef broth to prevent lumps, then add the Worcestershire sauce, garlic or onion powder, and pepper.

Return the hamburger steaks and any accumulated juices to the skillet. Cover and simmer gently for 10–15 minutes, turning the patties once, until the centers reach at least 160°F and the gravy thickens.

Taste the gravy and adjust the salt as needed. If using sour cream, remove the skillet from the heat and stir it in at the end; do not boil the gravy afterward.