Heat the broiler: Position an oven rack about 6 inches from the heat and turn the broiler to high. Set a 10-inch oven-safe skillet on the stove.

Prep the scallions: Slice the scallions if you have not already. Keep the white parts in one pile and the green parts in another.

Loosen the rice: Break up the cold rice with your fingers or a fork so there are no large clumps. This helps the rice heat evenly in the skillet.

Start the skillet: Put the skillet over medium heat and add the butter. When the butter melts, swirl the skillet so the bottom is coated.

Cook the aromatics: Add the scallion whites and garlic to the skillet. Cook for 1 minute, stirring often, until they smell fragrant and look slightly softened.

Cook the kimchi: Add the chopped kimchi and the reserved kimchi brine to the skillet. Cook for 2 to 3 minutes, stirring once or twice, until some of the liquid cooks away and the kimchi looks a little darker.

Bloom the chile paste: Add the gochujang to the skillet. Stir for about 30 seconds so it coats the kimchi and starts to loosen in the heat.

Add the rice: Add the rice to the skillet. Fold and stir for 1 to 2 minutes, breaking up any last clumps, until the rice is evenly mixed with the kimchi.

Season the rice: Pour the soy sauce over the rice. Stir for 2 minutes until the rice is hot all the way through and evenly seasoned.

Crisp the bottom: Use the back of your spoon to press the rice into an even layer in the skillet. Let it cook over medium heat for 2 to 3 minutes without stirring so a few crisp bits form on the bottom.

Finish the rice base: Drizzle the sesame oil over the rice. Taste a small spoonful and add a pinch of salt only if it needs it, since kimchi and soy sauce are already salty.

Add the cheese: Scatter the shredded mozzarella over the top of the rice. Leave four small open spaces for the eggs.