Let's cook.
In a medium bowl, whisk together the eggs, milk, salt, and black pepper until well combined and slightly frothy.
Heat the sesame oil in a non-stick skillet over medium heat until shimmering.
Add the chopped kimchi to the skillet and sauté for 2-3 minutes, stirring occasionally, 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 push the cooked egg from the edges towards the center with a spatula, allowing the uncooked egg to flow underneath. Continue this process for 2-3 minutes until the eggs are mostly set but still slightly moist.
Sprinkle the shredded cheese over the eggs. Cover the skillet with a lid for 1 minute to allow the cheese to melt.
Remove from heat, garnish with sliced green onions, and serve immediately.
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