01 - In a small bowl, combine gochujang, soy sauce, toasted sesame oil, sugar, and black pepper. Stir until smooth and set aside.
02 - Heat neutral oil in a large skillet or wok over medium-high heat until the oil shimmers.
03 - Add minced garlic, diced carrot if using, and the white parts of the scallions. Stir-fry for 1 minute until fragrant and softened.
04 - Add chopped kimchi to the wok and cook for 2 minutes, stirring frequently, until slightly caramelized and deeply fragrant.
05 - Add the cold rice to the wok, breaking apart any clumps with a spatula. Stir-fry for 2 to 3 minutes until the rice is heated through and slightly toasted.
06 - Pour the prepared sauce mixture and kimchi juice over the rice. Toss vigorously until every grain is evenly coated and the rice turns a vibrant reddish color.
07 - Fold in half of the scallion greens. Taste and adjust seasoning with additional soy sauce or pepper as needed.
08 - In a separate small pan, fry eggs to your preferred doneness, either sunny-side up or over-easy.
09 - Serve the fried rice hot, topped with fried eggs if desired, extra scallion greens, and a sprinkle of toasted sesame seeds.