Stir the soy sauce, brown sugar, and gochujang together in a small bowl.
Brown the beef in a wide pan over high heat. Spread it out and leave it alone for 2 to 3 minutes before breaking it up. That undisturbed contact is where the flavor comes from.
Add garlic and ginger, stir 1 minute, then pour in the sauce and let it bubble and reduce until it coats the meat, 2 to 3 minutes.
Kill the heat and stir through the sesame oil and most of the scallions.
Fry the eggs in a separate pan, keeping the yolks runny.
Serve the beef over rice, topped with an egg, the remaining scallions, and sesame seeds. Kimchi on the side if you have it.