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