In a large bowl, combine ground beef, panko, grated onion (and its juices), egg, garlic, Worcestershire, soy sauce, and black pepper. Mix with a fork or your hands just until evenly combined. Form 4 oval patties about 3/4 inch thick and press a shallow thumbprint in the center of each. Chill 10 minutes to help them hold their shape.
While the patties chill, heat 1 tablespoon olive oil in a large skillet over medium heat. Add mushrooms in an even layer and cook undisturbed for 4 minutes to brown. Stir in the sliced onions and a pinch of salt; cook 4 to 5 minutes more, stirring occasionally, until the onions are tender and mushrooms are well browned. Add the minced garlic and cook 30 seconds. Transfer veggies to a bowl.
Return the skillet to medium-high heat and add 1 tablespoon olive oil. Sear the patties until deeply browned, about 3 minutes on the first side and 2 to 3 minutes on the second. Transfer to a plate (they will finish cooking in the gravy).
Reduce heat to medium. Melt the butter in the same skillet. Sprinkle in the flour and stir constantly for 1 minute to form a blond roux.
Pour in the sherry and scrape up the browned bits. Gradually whisk in the beef broth until smooth. Whisk in Dijon, Worcestershire, ketchup, and dried thyme; bring to a gentle simmer.
Slide the mushrooms and onions back into the pan, then nestle in the patties along with any accumulated juices. Spoon some gravy over the tops. Cover loosely and simmer on medium-low for 7 to 9 minutes, or until the gravy is glossy and the patties reach 160°F in the center.
Uncover and simmer 1 additional minute if you want the gravy a touch thicker. Taste and season with salt and pepper as needed. Let rest off heat for 3 minutes before serving.
Serve the steaks with plenty of mushroom-onion gravy over mashed potatoes, rice, or buttered egg noodles.