To make the patty, combine the ground beef, chopped onion, garlic powder, egg, panko bread crumbs, Worcestershire sauce, salt, and pepper in a mixing bowl. Mix well until all the ingredients are thoroughly combined.
Shape the beef mixture into 4 patties, about ½ to ¾ inch thick.
Heat a skillet over medium heat and cook the beef patties for 4-5 minutes per side, or until they reach your desired level of doneness. Remove the patties from the skillet and set them aside.
For the gravy, melt the half stick of butter in the skillet (cleaned) over medium heat.
Add the chopped mushrooms to the skillet and sauté for about 5-6 minutes until they become tender.
Sprinkle the flour over the mushrooms and butter. Stir well to combine and cook for 1-2 minutes, allowing the flour to cook slightly. The mixture should look like a blond roux.
Gradually pour in the beef broth while stirring continuously to avoid lumps. Add the Worcestershire sauce and soy sauce to the skillet and mix well.
Bring the mixture to a simmer and cook for about 5 minutes, or until the gravy thickens. Stir occasionally. Season with salt and pepper to taste.
To assemble, divide the cooked jasmine rice among 4 plates. Place a beef patty on each bed of rice. Spoon the mushroom gravy over the beef patties, and top with sunny-side up eggs. Garnish with chopped green onions.