Grate the onion using the large holes of a box grater into a bowl. Keep the juices — that moisture keeps the patties tender.
In a large bowl, combine the ground beef, grated onion, panko, egg, garlic, Worcestershire, salt, and pepper. Mix with your hands until just combined, then stop.
Divide the mixture into 4 equal parts and shape into oval patties about 3/4-inch thick, slightly thinner in the center so they cook evenly.
Heat a drizzle of oil in a 12-inch skillet over medium-high heat until shimmering. Add the patties and cook without moving until deeply golden, 3-4 minutes per side. Transfer to a plate.
Reduce heat to medium and add the butter, sliced mushrooms, and onions. Cook until they release their liquid, sizzle away, and start to brown, 5-6 minutes, scraping up the browned bits from the pan bottom.
Sprinkle the flour over the vegetables and stir constantly for 1 full minute. It will look like a thick paste — that is exactly right.
Slowly pour in the beef broth while stirring constantly. Add the Worcestershire, Dijon, and thyme. Bring to a gentle simmer and let it bubble for 3-4 minutes until it coats the back of a spoon. Taste before adding salt.
Return the patties to the pan, spoon a little gravy over the tops, and let them cook gently until cooked through, about 5-6 minutes. Use a gentle simmer, not a hard boil.
Turn off the heat and rest 2-3 minutes. Spoon the gravy over the patties and serve over mashed potatoes or egg noodles.