Combine the panko and milk in a large bowl and rest for 5 minutes. Add the egg, grated onion, garlic, Worcestershire, Dijon, ketchup, salt, and pepper and stir well.
Add the ground beef and mix gently just until combined. Divide into four portions and shape into 3/4-inch-thick ovals. Press a shallow indentation in each center. Chill for 10 minutes if very soft.
Heat olive oil in a large skillet over medium-high. Sear the patties undisturbed for 3 to 4 minutes per side. Transfer to a plate; they will finish cooking in the gravy.
Reduce heat to medium. Add butter, mushrooms, and sliced onion. Cook for 7 to 9 minutes until the moisture evaporates and the onions begin to caramelize. Stir in thyme for 30 seconds.
Sprinkle flour over the vegetables and stir for 1 minute. Slowly whisk in beef broth while scraping up the browned bits. Add Worcestershire and Dijon. Simmer for 3 to 5 minutes until lightly thickened.
Return the patties and their juices to the skillet. Spoon gravy over the tops, cover, and simmer on low for 8 to 10 minutes until the centers reach 160°F (71°C). Add broth if the gravy becomes too thick.
Taste the gravy, season with salt and pepper as needed, garnish with parsley, and serve hot with mashed potatoes or noodles.