Shred about a cup of Green Cabbage and grate one Carrot; finely dice half a Onion and thinly slice a few Scallion. Put the vegetables in a bowl and set aside.
Crack 1–2 Egg into a separate bowl and beat lightly, then stir in the shredded cabbage, carrot, onion and scallion. Season with a pinch of Fine Salt, a grind of Black Pepper and a small pinch of Granulated Sugar to mimic the slightly sweet profile of Korean toast.
Heat a nonstick skillet over medium with a teaspoon of Neutral Oil. Pour the egg-vegetable mixture in and spread it out into a thin, even pancake; cook until set and lightly golden on both sides, then remove and keep warm.
Spread Unsalted Butter on one side of each slice of Soft White Sandwich Bread. In the same skillet over medium heat, toast the buttered sides until golden and crisp.
On the untoasted sides of the bread, spread a thin layer of Mayonnaise, a little Ketchup and a touch of Yellow Mustard on one slice (adjust ratios to taste).
Assemble the sandwich by placing the egg-vegetable patty on the sauced slice, then top with a slice of Ham and a slice of American Cheese. Close with the second slice of bread, toasted-side out.
Return the sandwich to the skillet and press gently with a spatula, cooking briefly on each side until the cheese softens and the bread is golden and crisp. Slice in half and serve warm.
