Egg toast is a ubiquitous Korean street breakfast, providing a nutritious and filling start to the day. The cheese and ham are lovely optional additions, but the brown sugar, unusual though it may seem, is pretty much essential to the egg toast experience.
Whisk the eggs in a small bowl, then add the carrot and cabbage and mix well. Season with a little salt and pepper. Set aside. Melt half the butter in a small, preferably non-stick, frying pan. When it is foaming, add the slices of bread and, pressing down with a spatula, fry them until crisp. Turn over and fry on the other side until crisp. Remove to a plate and keep warm.
Add the rest of the butter to the pan and pour in the egg mixture, spreading it out evenly to an approximately toast-sized omelette. Cook over a medium-low heat for a few minutes until set, then use a fish slice to flip over. Lay the ham and cheese slices on top, if using, so that they warm a little, and cook for another minute until the egg is lightly crisp on the underside.
Lift the egg, ham and cheese on to one of the slices of toasted bread. Sprinkle over the brown sugar, followed by a good squirt of ketchup and chilli sauce, before topping with the other piece of toast. Cut in half if you like, or just grab it whole, and eat immediately.