Step 1

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.

Step 2

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.

Step 3

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.

Step 4

Stack cabbage pieces on top of one another and transfer to toast. Top with sugar.

Step 5

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.

Step 6

Place ham on sandwich and squeeze over mayonnaise and ketchup. Top with remaining slice of toast, slice in half, and serve.