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1
Using a fork, beat the eggs in a small bowl until there are no visible bits of white or yolk (you're just trying to get an even mixture, not incorporate a ton of air) and season them with salt and pepper.
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2
Heat the butter in a medium skillet, preferably carbon steel, nonstick, or well-seasoned cast iron, over medium-high heat. Once the butter is melted and foamy, add the eggs, swirling the pan to make sure they are forming an even layer. Immediately sprinkle the eggs with the cheese, followed by the kimchi. Let the eggs cook until the underside is browned and slightly puffed but the top is still rather runny, 2 to 3 minutes.
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3
Using a spatula (preferably silicone if using a nonstick pan), lift the edge of the eggs all around the skillet to release them and, starting at one end, fold the omelette onto itself. You can either roll it like a classic omelette or, if that's just not an option this morning, simply fold it into a half-moon.
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4
NOTE: I prefer my omelettes with two eggs rather than three, which means they'll have an almost crepe-like thinness, but if you like a bit more heft, or are especially hungry, you can certainly use three. I use a carbon-steel skillet to make my omelettes, but nonstick or even a well-seasoned cast-iron skillet will work as well.
