Mix ground beef, onion, garlic, soy sauce, Worcestershire sauce, salt, pepper, egg, and breadcrumbs just until combined.
Shape the mixture into 4 patties.
Cook the patties in a skillet over medium heat for 4–5 minutes per side, then remove and set aside.
Melt the butter in the same skillet.
Whisk in the flour and cook for 1 minute until lightly golden.
Slowly pour in the beef broth while whisking to prevent lumps.
Stir in soy sauce and Worcestershire sauce, then simmer 4–6 minutes until the gravy thickens.
Season the gravy with salt and pepper.
Fry the eggs sunny-side-up or over-easy.
Scoop rice onto plates, place a patty on top, pour gravy over it, and finish with a fried egg.
Add sliced green onions if desired.