⅓ cup julienned carrot
1 cup thinly sliced green cabbage
1 stalk green onion, thinly sliced
¼ teaspoon ground black pepper
teaspoon kosher salt, or to taste
2 large eggs
3 tablespoons salted butter,
2 thick slices white bread
1 teaspoon white sugar, or to taste
2 thin slices ham
1 American cheese
Combine carrot, cabbage, onion, black pepper, , 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 pan 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 pan 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 pan over medium heat; cook until sizzling. Flip and top ham with cheese. Cook until melted, about 1 minute.
Place ham on sandwich and squeeze over mayonnaise and ketchup. Top with remaining slice of toast.