Break up the chilled rice with your hands or a fork so there are no large clumps. Stir the gochujang, soy sauce, and 2 tablespoons of kimchi juice together in a small bowl and set it near the stove.

Heat 1 tablespoon of the oil in a large skillet or wok over medium-high heat. Add the chopped kimchi and cook, stirring often, 3 to 4 minutes, until softened and slightly darkened. If your kimchi is very sour, sprinkle in the sugar now.

Push the kimchi to one side, add the garlic and scallion whites to the empty space, and cook about 30 seconds, until fragrant. Stir everything together.

Add the rice and pour the sauce mixture over the top. Toss and press the rice into the pan, then let it sit undisturbed 1 to 2 minutes so the bottom grains toast. Toss and repeat once more. Add the peas or carrots, if using, with the rice.

Taste and adjust: add the remaining kimchi juice for tang, more soy sauce for salt, or more gochujang for heat. Cook 1 to 2 minutes longer, until hot through with slightly chewy edges.

Turn off the heat, drizzle the sesame oil over the rice, and toss once more. Shower with most of the scallion greens and the sesame seeds.

For the eggs, wipe out the pan, heat the remaining tablespoon of oil over medium heat, and fry the eggs 2 to 3 minutes, until the whites are set and the edges crisp. Serve the rice in bowls topped with an egg, more scallion greens, sesame seeds, and crumbled seaweed if you like.