Combine carrot, cabbage, green onion, black pepper, cayenne, and salt. Mix with your hands for about 30 seconds until soft and seasoned. Add eggs and mix thoroughly with a fork; set aside.
Heat 2 tablespoons butter in a nonstick skillet over medium-high heat. Add bread slices and toss to make sure both sides are well buttered. Grill bread until both sides are beautifully toasted and golden brown and starting to get a little crispy. Remove to a plate.
Heat remaining butter in the same skillet over medium heat. Add cabbage mixture; cook and simultaneously mix by using the edge of a spatula to chop down on the surface while pushing the edges up and over the top. As the mixture starts to hold its shape, form into a rectangle that’s about twice as long as it is wide. Split the rectangle in half and continue to cook until golden brown; flip and continue on the opposite side, 3 to 4 minutes total.
Stack cabbage pieces on top of one another and transfer to toast. Top with sugar.
Add ham to the same skillet over medium heat; cook until sizzling. Flip and top ham with Cheddar cheese. Cook until melted, about 1 minute.
Place ham on sandwich and squeeze over mayonnaise and ketchup. Top with remaining slice of toast, slice in half, and serve.
You can grate the carrot or use a vegetable cutter instead.
You can use a pinch of Korean chile flakes instead of cayenne. You can use American cheese instead of mild Cheddar.
If you want to use fewer vegetables and 1 egg, you can make a smaller version of the same sandwich.
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