Make the Burger Patties: In a large mixing bowl, combine the ground beef, finely diced onion, panko breadcrumbs, egg, Worcestershire sauce, soy sauce, garlic powder, salt, and pepper.
Mix gently with your hands or a spoon until just combined—be careful not to overmix. Divide the mixture into four equal portions and form into wide, flat patties (they will shrink slightly as they cook).
Cook the Patties: Heat a large skillet over medium-high heat and add one tablespoon of oil. Sear the patties for 3-4 minutes per side, or until they reach your preferred doneness. Remove from the pan and set aside.
Make the Gravy: In the same pan, add the sliced mushrooms and cook for about 5-6, or until browned. Add the sliced onions and sauté for another 7-8 minutes until just caramelized. Stir in the butter and additional oil if needed. Sprinkle in the flour and stir for one to two minutes to cook out the raw flavor.
Slowly whisk in the beef stock, stirring constantly to avoid lumps. Simmer the gravy for 5-7 minutes, or until thickened. Season with Worcestershire sauce, soy sauce, salt, and pepper to taste.
Fry the Eggs: In a clean skillet, fry the eggs over medium heat until the whites are set but the yolks are still runny, about two to three minutes.
Assemble and Serve: Scoop the rice onto plates or bowls. Top each serving with a cooked burger patty, spoon over the mushroom onion gravy, and finish with a fried egg. Garnish with furikake and green onions, if desired.