In a large mixing bowl, add cabbage, onion, carrot, eggs, salt and pepper; combine well using chopsticks as the mixture will be to thick to whisk.
Heat a large skillet over medium heat; melt 2 tablespoons butter. Pour egg mixture, trying to make it a square shape. Cook until bottom is cooked, about 2 to 3 minutes. Cut egg in half, and flip over. Cook another 2 to 3 minutes until fully cooked.
Meanwhile butter, both sides of bread.
When egg mixture is cooked, remove from skillet and add ham slices to heat them up, after flipping once, place one slice of cheese over each slice.
Now, toast the buttered bread both sides, about 2 minutes each side, or until golden brown. Remove form skillet.
To build, Place egg on toast then place warm ham and cheese, lightly squirt the ketchup and sprinkle the sugar right on top. Cover with the other slice of buttered and toasted bread. enjoy!