Spread butter on both sides of the bread slices and keep them aside.
Combine the egg, cabbage, carrots, and onion in a jug/mixing bowl. Add salt and freshly ground black pepper and mix everything well.
Heat a pan over medium heat. Add 1 tbsp butter to the pan then pour in the egg mixture, try to give it a rectangular shape. Cook until the bottom is fully cooked, about 2 to 3 minutes. Then cut the omelet into a half and flip over. Cook the other sides well about 2 to 3 minutes or until fully cooked. Take it off the pan when both sides are fully cooked.
Now toast the buttered bread slices on the same pan (about 1-2 minutes each side or until golden brown)
Once both sides of the bread are toasted and golden brown, place cheese slice and the omelet into one bread slice, then sprinkle 1 tsp sugar on top of the omelet and add ketchup and mustard right on top (you can adjust the amount of the sugar, ketchup, and mustard according to your preference). Cover with another slice of toasted bread. Repeat the same process for the rest of the bread slices.
Slice the sandwich in the middle and serve.