In a bowl, whisk eggs. Add shredded cabbage, julienned carrot, finely diced onion, salt, and black pepper. Mix well.
Heat 1/2 tbsp of butter in a non-stick pan over medium heat. Fry ham slices until lightly browned. Remove from pan and set aside.
Using the same pan, add another 1/2 tbsp of butter. Pour half of the egg mixture into the pan and spread it into a rectangular shape, similar to the bread slice. Cook for 2-3 minutes per side until golden brown and cooked through. Repeat with the remaining egg mixture to make a second patty.
In the same pan, add the remaining 1 tbsp of butter. Toast the bread slices on both sides until golden brown and crispy.
Place one toasted bread slice on a plate. Layer with one egg patty, two slices of ham, and then sprinkle with sugar. Drizzle with ketchup and mayonnaise if desired. Top with the second toasted bread slice.
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