Let's cook.
In a medium bowl, whisk together the eggs, soy sauce, and black pepper until well combined.
Heat the sesame oil in a non-stick skillet over medium heat.
Add the chopped kimchi to the skillet and sauté for 2-3 minutes, until slightly softened and fragrant.
Pour the egg mixture into the skillet with the kimchi. Let it set for about 30 seconds without stirring.
Gently scramble the eggs with a spatula, breaking them into desired curds. Cook for another 2-3 minutes, until the eggs are cooked through but still moist.
Divide the warm cooked rice between two bowls.
Spoon the kimchi and egg scramble over the rice.
Garnish with sliced green onion and serve immediately.
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