Bring rice and water to a boil in a saucepan. Cover, reduce heat to low and cook for 18 minutes, then rest covered off heat for 10 minutes. Spread rice no more than 1 inch deep on a clean tray and refrigerate promptly, uncovered, for at least 1 hour until cold; include this chilling time in preparation.
Heat 1 tablespoon neutral oil in a wide skillet over medium-high heat. Brown beef for 7 to 8 minutes, breaking it into small crumbles and cooking to 160°F; transfer to a bowl.
Heat another tablespoon oil in the skillet. Add kimchi and garlic and cook for 4 to 5 minutes until the cabbage softens and visible liquid evaporates.
Mix kimchi juice, gochujang and soy sauce. Add cold rice and the sauce to the skillet, separate clumps with a spatula and fry for 5 to 6 minutes, turning every minute.
Return beef to the skillet and cook for 2 minutes until rice and beef are steaming throughout. Stir in sesame oil and scallions.
Heat the last tablespoon neutral oil in a separate nonstick skillet. Fry eggs for 3 to 4 minutes until the whites are set, covering briefly if needed; cook yolks firm if preferred. Divide rice among four bowls and put one egg on each.