Break up the cold rice with clean hands or a spoon so there are no large clumps. Chop the kimchi and reserve its brine.
Heat 2 teaspoons (10 mL) of the neutral oil in a large nonstick skillet over medium-high heat. Add the onion and cook, stirring often, until softened and lightly golden, about 3 to 4 minutes. Add the garlic and cook for 30 seconds, just until fragrant.
Add the chopped kimchi and cook for 2 minutes, stirring, until hot and slightly concentrated. Stir in the gochujang, soy sauce, and kimchi brine until the gochujang is dissolved.
Add the cold rice. Press and toss it through the kimchi mixture until evenly colored and heated through, about 4 to 5 minutes. Let it sit undisturbed for short intervals so a few grains can lightly crisp. Remove from the heat and stir in the sesame oil and most of the scallions.
While the rice cooks, heat the remaining 1 teaspoon (5 mL) neutral oil in a separate nonstick skillet over medium heat. Crack in the eggs and cook until the whites are set and the yolks are cooked to your preference, about 3 to 4 minutes for a runny yolk.
Divide the kimchi fried rice among bowls. Top each portion with a fried egg, remaining scallions, sesame seeds, and crumbled roasted seaweed, if using. Serve hot.