Step 1

Thinly slice 4 medium scallions (scant 1 cup); reserve about 1 tablespoon of the dark green parts for garnish. Finely chop 1 cup drained Napa cabbage kimchi into bite-sized pieces. Place 1 tablespoon mayonnaise, 1 tablespoon kimchi brine, 1 tablespoon granulated sugar, 1 tablespoon gochugaru, and 1 tablespoon plus 1 1/2 teaspoons soy sauce in a small bowl.

Step 2

Melt 2 tablespoons unsalted butter in a wok or large nonstick frying pan over medium-high heat. Add the scallions and cook, stirring occasionally, until fragrant, 1 to 2 minutes. Add the kimchi and cook until the kimchi softens, 2 to 3 minutes. Add the mayonnaise mixture and cook, stirring occasionally, until the kimchi and scallions absorb all the seasonings, 2 to 4 minutes.

Step 3

Add 2 cups cold cooked white rice, breaking up any big clumps with your hands as you add it. Stir-fry until combined. Turn off the heat. Season with 1 tablespoon toasted sesame oil and 1/4 teaspoon kosher salt, and toss to combine. Taste and season with more kosher salt as needed.

Step 4

Heat 1 tablespoon neutral oil in a medium nonstick frying pan over medium-high heat until shimmering. Add 2 large eggs and fry until the whites are set but the yolks are still runny, 1 1/2 to 2 minutes. Take the pan off the heat. Using the edge of a flat spatula, cut through the 2 fried eggs to separate if needed.

Step 5

Divide the fried rice between 2 bowls. Top each bowl with a fried egg. Garnish with the reserved scallions and toasted sesame seeds.