Thinly shred the white cabbage and julienne or finely dice the carrot. Set aside.

In a medium bowl, whisk the egg. Add the shredded cabbage, diced carrot, 1/2 teaspoon of the sugar, 1/2 teaspoon soy sauce, 1/4 teaspoon sesame oil (if using), a pinch of salt, and pepper. Mix everything well.

Heat a non-stick pan or griddle over medium heat. Add a small amount of oil or butter. Pour the egg and vegetable mixture into the pan, shaping it into a rectangular patty that fits your bread slices. Cook for 3-4 minutes per side, or until golden brown and cooked through. Remove and set aside.

In the same pan, quickly fry the ham slices until lightly browned, about 1-2 minutes per side. Remove and set aside.

Lightly butter both sides of the bread slices. Toast them in the pan over medium heat until golden brown and crispy on both sides, about 1-2 minutes per side.

To assemble your Gilgeori Toast: Spread Kewpie mayonnaise generously on one side of a toasted bread slice. Place the cooked egg and vegetable patty on top. Add the fried ham slices. Drizzle generously with ketchup, then add another layer of Kewpie mayonnaise. Sprinkle the remaining 1/2 teaspoon of sugar over the fillings (this adds the signature sweet touch!). Cover with the second slice of toasted bread.

Serve your homemade Gilgeori Toast immediately and enjoy the authentic flavors!