Make egg mixture. In a large bowl, add egg, salt, and pepper. Whisk well with a fork. Add shredded cabbage, julienned carrot, and chopped green onion. Mix well until the egg coats all the veggies.

Toast bread. Heat skillet to medium heat. Melt 1 Tbsp butter and toast bread on one side. If needed, add 1 more Tbsp of butter and flip, toasting bread on other side. Set aside on a wire rack to cool slightly or tent bread against each other so the steam can escape.

Cook egg mixture. In the same skillet, carefully add eggy mixture in a square-ish mound about the same size as the bread. Using a metal spatula, shape the eggy filling into a square. Cook until the eggy edges look firm and the veggies have wilted a bit, about 3-4 minutes. Flip and cook until the mixture is completely cooked through, another 3-4 minutes. Lower heat to medium low, if the egg is burning or cooking too quickly.

Add sugar. Turn off the heat. Add 1-2 tsp sugar (to taste) directly over the cooked egg mixture.

Add ham and cheese. Add ham and cheese on top of the cooked egg mixture. If you prefer, briefly heat the ham on the skillet before transferring to the vegetable mixture.

Add mayo and ketchup or sriracha. Be generous!

Assemble. Make the sandwich, starting with one slice of buttered toasted bread then the egg mixture topped with ham and cheese. Add more cabbage on top, if desired, for added crunch and freshness. Top with remaining slice of bread. Cut in half and enjoy!