Whisk together the stock, soy sauce, Worcestershire, cornstarch, and ketchup in a bowl.
Melt 1 teaspoon of butter in a skillet over medium-high heat. Cook patties in the hot skillet until the bottoms are browned. This takes about 5 minutes. Turn patties. Add green onions and cook until both beef and onions are browned, 4 to 5 minutes. It usually takes less time on the second side. Remove the patties and green onions to a plate.
Pour the beef broth mixture into the skillet. Stir and scrape the browned bits from the bottom. Stir until thick, about 2 minutes. Reduce heat to low and reserve until needed.
Melt the remaining butter in another skillet over medium heat. Add the eggs and cook until the whites are set and the yolks are still runny. (Or, to your liking)
Divide rice between 2 serving bowls. Place beef over rice, spoon gravy on top, and top with your fried egg.