Beat eggs in a large mixing bowl and add the shredded cabbage, carrot, green onion, and a pinch of salt; mix well.
Melt 1 tablespoon of butter on a skillet over medium low heat. Spoon in a portion of egg and vegetable mixture, and spread it around to about the size of your sandwich bread. (You can make two egg pieces at a time)
Cook the egg omelette 2-3 minutes on one side and flip. Cook until golden brown. Transfer the omelette to a plate and repeat the next batch.
Heat ham slices briefly, about 30-60 seconds; set aside.
Melt remaining tablespoon of butter and toast the sandwich bread until golden and crisp.
To assemble the toast
Put a piece of egg omelet on the toast. Sprinkle about 2 teaspoon of sugar (or however much you desire), and squirt some ketchup on top. Place a slice of ham and a cheese on top and top with the another toasted bread.
If you like your toast crunchier, add shredded cabbage between the layers of the condiments and ham.