Korean Toast Sandwich

Combine the shredded cabbage, peeled carrot, chopped green onion, salt, pepper, and a pinch of heat with your hands in a bowl until the ingredients are well incorporated and soft, approximately 30 seconds.

Combine the eggs with a fork until they are fully merged and combined.

Heat 2 tablespoons of butter in a non-stick skillet over medium heat. Add slices of bread and gently flip them several times to ensure both sides are evenly coated with butter. Cook the bread until it is perfectly toasted, golden brown, and has a slight crunch on both sides. Transfer the cooked bread to a plate.

Melt the remaining 1 tablespoon of butter in the same skillet over medium heat. Add the cabbage mixture and cook, stirring occasionally with the edge of a spatula to gently break down the ingredients on the surface while pushing the edges up and over the top. As the mixture begins to hold its shape, shape it into a rectangular form that is roughly twice as long as it is wide. Divide the rectangle in half to create two square patties; continue to cook until they are golden brown, 1 1/2 to 2 minutes per side.

Stack the patties on top of one another onto a slice of toast and sprinkle with sugar.

Add slices of ham to the hot skillet; cook over medium heat until sizzling, 1 to 2 minutes. Flip and place a slice of Cheddar cheese on top; cook until the cheese is melted, about 1 minute.

Place the ham and cheese on top of the patties; place a dollop of ketchup, mayonnaise, and remaining slice of toast on top. Cut in half and serve.