Combine the rice, water, and salt in a saucepan. Bring to a boil, cover, reduce the heat, and cook for 15 minutes. Rest off the heat for 10 minutes, then fluff.
Cook the Patties:
Mix the beef, grated onion, soy sauce, Worcestershire sauce, salt, and pepper just until combined. Shape into four wide patties.
Cook in a hot skillet over medium-high heat for 4 to 5 minutes per side, or until cooked to preference. Transfer to a plate and keep warm.
Make the Gravy:
Melt the butter in the same skillet. Cook the mushrooms and sliced onion for 7 to 8 minutes until browned.
Stir in the flour for 1 minute. Gradually whisk in the stock, soy sauce, and Worcestershire sauce. Simmer for 5 minutes until glossy and thickened.
Fry and Serve:
Heat the oil in a clean skillet and fry the eggs until the whites are set and the yolks remain runny.
Divide the rice among plates, top with patties, gravy, and fried eggs, and add macaroni salad on the side.
Notes
Do not overmix the beef or the patties may become dense.
For food safety, cook ground beef to an internal temperature of 160°F (71°C).