Remove the chilled rice from the refrigerator and break apart any large clumps with clean fingers or a fork. The grains should be loose enough to spread easily through the skillet.
In a small bowl, stir together the kimchi juice, gochujang, and soy sauce until the gochujang is mostly dissolved.
Heat 1 tablespoon (15 ml) neutral oil in a 12-inch wok or large skillet over medium-high heat until shimmering.
Add the diced onion and white parts of the green onions. Cook for 2 to 3 minutes, stirring frequently, until the onion is softened and lightly translucent.
Stir in the minced garlic and cook for about 30 seconds, until fragrant but not browned.
Add the chopped kimchi and stir-fry for 3 to 4 minutes. Some moisture should evaporate, and the edges of the kimchi should begin to soften and lightly caramelize.
Add the chilled rice and remaining 1 tablespoon (15 ml) neutral oil. Toss and press the rice through the kimchi mixture until the grains are separated and evenly distributed.
Pour the kimchi juice, gochujang, and soy sauce mixture over the rice. Toss thoroughly until the grains are evenly coated and turn reddish-orange.
Spread the rice into an even layer and let it cook undisturbed for 1 to 2 minutes. Toss, spread again, and cook for another 1 to 2 minutes. The rice should be steaming hot with a few lightly browned, crisp areas.
Remove the skillet from direct heat: Stir in the toasted sesame oil and most of the green onion tops, reserving a small amount for garnish.
Heat 1 teaspoon (5 ml) neutral oil in a nonstick skillet over medium-low heat. Crack in the eggs and cook until the whites are fully set and the yolks reach your preferred doneness. For fully cooked eggs, cook until the yolks reach 160°F (71°C).
Let the fried rice stand off the heat for 5 minutes. The grains will settle slightly while remaining hot.
Divide the Kimchi Bokkeumbap among four bowls or plates and top each serving with one fried egg.
Sprinkle with toasted sesame seeds, roasted seaweed strips, and the reserved green onions as desired. Serve immediately.