Prepare the patties. Combine ground beef with soy sauce, Worcestershire, salt, pepper, chopped green onion, and garlic. Form into 4 equal patties about ½ inch thick.
Heat a large saute pan over medium heat. Add patties and cook until browned on both sides, about 3 minutes. Remove from heat and set aside.
In the same pan, heat 1 tablespoon of oil. Add sliced onions and cook until browned, about 5 minutes. Reduce heat to medium-low and cook until golden brown and caramelized, about 20 minutes. Add 1-2 tablespoons of water to the pan if it gets too dry.
Move caramelized onions to the edges of the pan and add mushrooms. Continue to saute until browned and season with salt and pepper.
Pour stock and soy sauce to the pan, and bring to a simmer. Continue to simmer for 5 minutes. Add cornstarch slurry and simmer for another minute or until slightly thickened. Season gravy with salt and pepper.
Add beef patties to gravy, spooning gravy on top. Simmer for 2 minutes or until warm. Plate 1 cup rice on each plate, topped with 1 patty, gravy with mushrooms and onions, and a fried egg. Serve immediately.
Notes
The gravy can be prepared the day before and reheated before serving.