Marinate the Beef: In a bowl, combine the thinly sliced beef with cornstarch, sesame oil, soy sauce, sugar, and Shaoxing wine until evenly coated, then set it aside to marinate for at least 15 minutes.
Cook the Eggs: Just before preparing the fried rice, cook the 4 eggs sunny-side up in a pan, keeping the yolks runny, then keep the eggs warm while you cook the rice.
Heat the Wok: Place a wok over high heat and add 2 tablespoons of the oil, allowing it to heat until the wok begins to smoke.
Sear the Beef: Add the marinated beef to the smoking-hot wok and sear it, then transfer the cooked beef to a plate and set aside.
Cook the Onion: Add the remaining 1 tablespoon oil to the wok, then stir-fry the finely chopped onion until it becomes translucent.
Add the Pepper: Add the chopped red bell pepper and cook for another 1 minute, stirring as it cooks.
Add the Kimchi: Stir the coarsely chopped kimchi into the onion and pepper mixture, combining everything thoroughly.
Fry the Rice: Add the cooked white or brown rice to the wok and stir constantly, mixing it thoroughly with the kimchi and vegetables.
Return the Beef: Add the seared beef back into the wok and toss it through the rice mixture.
Season the Rice: Add salt to taste, ½ teaspoon sesame oil, the chopped scallions, and a pinch of Korean chili flakes if desired for additional heat.
Finish the Fried Rice: Thoroughly mix all the ingredients together so the beef, vegetables, kimchi, seasonings, and rice are evenly distributed.
Serve: Divide the fried rice among servings and top with the warm sunny-side-up eggs, keeping the yolks runny for a luscious finish.
