Rinse medium-grain rice under cold water until water runs clear. Combine rinsed rice with water in a rice cooker. Cook until tender and fluffy, about 20 minutes.

In a medium saucepan, combine beef stock, cornstarch, onion powder, shoyu, and beef bouillon paste. Heat over medium, whisking until mixture thickens, about 5-7 minutes. Stir in heavy cream and simmer.

In a bowl, combine ground beef with egg, diced onion, panko, ketchup, mayonnaise, salt, and pepper. Shape into patties, about ½ inch thick. Cook in a skillet with oil for 4-5 minutes per side until browned.

In the same skillet, add more oil if needed. Crack eggs and cook gently for 3-4 minutes until whites are set but yolks are runny. Flip carefully and remove when cooked.

Plate rice, top with a patty, pour gravy over patty, and add a fried egg on top. Serve immediately.