Shred the cabbage, julienne the carrots, and finely chop the green onion; set aside. In a small bowl, beat the eggs, then add the prepared vegetables and mix well.

Optional: On a frying pan, lightly toast the ham slices for about 30 seconds on each side.

Preheat a nonstick pan over medium heat and melt a little butter. Pour in the egg mixture, spreading it into a long rectangle wide enough to fit two slices of bread. Cook until the bottom is lightly golden and the top is just set, no longer jiggly but still slightly soft.

Place the two slices of bread on top of the egg, positioning them so the bottom edges touch in the center and the top edges face outward. Once the egg is fully cooked, add a bit more butter to the pan, then flip the egg and bread together so the bread is on the bottom and the egg is on top.

Swirl the bread around the pan to toast it in the butter until golden. Turn off the heat, top with (optioanl) sugar, cheese slice and ham, then spread ketchup and butter as desired. Fold the other slice of bread over the fillings. Enjoy!