Let's cook.
In a shallow bowl, whisk together the 2 large eggs, 1/2 teaspoon sugar, and 1/4 teaspoon black pepper until well combined.
Spread 1/2 tablespoon of mayonnaise evenly on one side of each of the 4 slices of bread.
Dip each slice of bread into the egg mixture, ensuring both sides are coated. Do not let the bread soak too long, just enough to coat.
Heat 1 tablespoon of butter in a large non-stick skillet over medium heat. Once melted, place the egg-coated bread slices into the skillet.
Cook for 2-3 minutes per side, or until golden brown and the egg is set. While the second side is cooking, sprinkle 1/4 cup of chopped kimchi and 1/4 cup of shredded mozzarella cheese over two of the bread slices.
Once the cheese is melted and the bread is golden, place the plain toasted bread slices on top of the kimchi and cheese slices to create two sandwiches. Press gently.
Remove from heat, slice in half, and serve immediately.
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