Bring the rice and water to a boil in a saucepan, cover and cook on low for 15 minutes, then rest off the heat for 5 minutes.
Shape the beef into four patties about 1/2 inch thick. Season with 1/2 teaspoon salt and the pepper; sear in a large skillet over medium-high heat for 4 minutes per side, until the centers reach 160°F, then transfer to a plate.
Leave 1 tablespoon drippings in the skillet. Add the onion and butter and cook for 3 minutes, then stir in flour for 30 seconds. Slowly stir in broth, soy sauce and Worcestershire; simmer for 3 minutes until lightly thickened.
Meanwhile, heat the oil in a second skillet, add the eggs and remaining salt, cover and cook over medium-low heat until whites and yolks are set.
Divide the rice among plates, add a patty to each and spoon the onion gravy over it. Top with the cooked eggs and serve.