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Season patties with salt, pepper, and cayenne; set aside.
Whisk together beef stock, soy sauce, Worcestershire sauce, cornstarch, ketchup, sugar, and sesame oil in a bowl; set aside.
Melt 1 teaspoon butter in a skillet over medium-high heat. Cook patties in the hot skillet until the bottoms are nicely browned, about 5 minutes. Flip patties; add green onions and cook until both beef and onions are browned, 4 to 5 minutes. Remove patties and most of green onions to a plate.
Pour beef stock mixture into the skillet. Stir, scraping up browned bits from the bottom, until thickened, about 2 minutes. Reduce heat to low and reserve gravy until needed.
Melt remaining butter in another skillet over medium heat. Add eggs to the skillet and fry side-by-side until whites are set and yolks are still runny.
Divide rice between 2 serving bowls. Place patties over rice, spoon gravy on top, and add fried eggs.
Serve hot and enjoy!