Rinse the rice under cool water. Place the rinsed rice and 2 cups water in a rice cooker and cook.
Shape the ground beef into four patties. Season both sides with salt and pepper.
Heat olive oil in a cast iron skillet over medium-high heat. Add the patties. Sear for 4 minutes. Flip. Cook for 4 more minutes. Remove the patties and set aside.
Reduce heat to medium-low. Add the flour to the skillet. Cook for 1-2 minutes, stirring continuously.
Add 1/2 cup beef broth to the skillet, and stir to combine.
Add the remaining beef broth and soy sauce. Cook until thickened, about 5 minutes. Whisk in butter. Season with salt and pepper to taste.
In another skillet, heat 2 teaspoons olive oil over medium-high high. Crack the eggs into the pan and fry for 3-4 minutes, until the whites are cooked through. The yolks should remain loose.
Place a large circle cookie cutter on a plate. Fill with a spoonful of rice. Remove the cookie cutter.
Place a cooked patty on top of the rice. Spoon on a ladle of gravy. Top with a cooked egg, and garnish with green onions.