Step 1

Season patties with salt, pepper, and cayenne.

Step 2

Whisk beef stock, soy sauce, Worcestershire sauce, sesame oil, ketchup, sugar, and cornstarch together in a bowl.

Step 3

Melt butter in a skillet over medium-high heat. Cook patty in the hot skillet until a browned crust forms on the bottom, about 5 minutes. Flip; toss in green onions and cook the other side until both beef and onions are browned, 4 to 5 minutes more. Remove patty and most of the green onions to a plate.

Step 4

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.

Step 5

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.

Step 6

Divide rice between 2 serving bowls. Place patties over the rice, spoon gravy on top, and add the fried eggs.