Make the patties: Combine the ground beef, egg, panko, Worcestershire sauce, garlic powder, salt, and pepper in a large bowl. Mix gently until combined. Shape the mixture into 4 equal patties. Make an indention in the center of each patty to prevent them from shrinking.
Heat the oil and butter in a large skillet over medium-high heat until the butter melts. Add the patties and cook until browned on each side, about 2 to 3 minutes per side. Remove the patties to a plate. Drain the oil, reserving 1 tablespoon along with the browned bits in the bottom of the pan.
Start the gravy: Add the onion and mushrooms to the drippings in the skillet. Cook, stirring occasionally, until the onions and mushrooms are browned and tender, about 6 to 8 minutes.
Whisk together the beef broth, cornstarch, soy sauce, and tomato paste in a liquid measuring cup. Add the beef broth mixture to the onion and mushrooms in the skillet and bring to a boil. Reduce the heat and simmer, stirring frequently, until the gravy is thickened. Return the patties to the skillet and cover in the gravy.
To serve, divide the rice evenly between serving bowls or plates. Top each with a hamburger patty and spoon gravy over the top. Top with a fried egg and sprinkle with green onions.