3/4cup cored, thinly sliced green cabbage (about 1 1/2 ounces)
1/3cup julienned or shredded carrot
1/4cup thinly sliced onion or shallot
1 large scallion, white part thinly sliced and green part cut into 1-inch lengths
Salt and pepper
1 large egg
2 slices hearty sandwich bread of choice
1 1/2 tablespoons butter, softened, divided
Sugar
Ketchup, to taste, optional
Mayonnaise, to taste, optional
1 or 2 thin slices deli or smoked ham, optional
1 or 2 thin slices good melting cheese, optional
Instructions
Add the cabbage, carrot and onion to a medium bowl and briefly knead with your hands to soften it. Add a big pinch of salt, pepper to taste, and the scallion, stir to distribute, and scrape the veggies to one side of the bowl, leaving the other side mostly open. Crack the egg into the open area, add a pinch of each salt and pepper, beat to combine the yolk and the white, and then mix it into the veggies to combine thoroughly.
Set a medium stick-resistant skillet over medium heat and allow it to heat for a minute or two. Meanwhile, spread one side of each bread slice with 1 1/2 teaspoons of the butter, spreading it all the way to the edges. Add the bread slices to the skillet buttered side down and cook, undisturbed, until golden brown, toasted, and starting to crisp on the bottom, about 2 minutes. Flip the slices and continue to cook, undisturbed, until dry and beginning to toast on the bottom, about 1 1/2 minutes longer. Transfer the bread to a plate, browned sides down.
Return the skillet to medium heat, add the remaining 1 1/2 teaspoons of butter, allow it to melt and bubble, and tilt the skillet to cover most of the center (an area about the size of your outstretched hand). Add the cabbage mixture and, using a wide, thin spatula, pat and press the surface and straighten the edges to shape it into a patty roughly the size and shape of your bread slices. Cook, continuing to straighten the edges as necessary (once the egg sets the patty should hold its shape), until browned on the bottom, about two minutes. Flip the patty, nudge it back into shape if necessary, and continue cooking until spotty brown on the second side, about 2 minutes longer. Set the patty on one of the toasted bread slices, sprinkle it evenly with a tiny pinch of sugar, and add ketchup and mayo, if desired. If not using ham and/or cheese, skip to step 5.
Meanwhile, if using the ham, return the skillet to medium heat, add the ham, and cook, undisturbed, until it sizzles slightly, about 1 1/2 minutes. Flip it over, and, if using cheese, place it on the ham; cook, undisturbed, until warmed through, about 1 minute longer. Place the ham (or ham and cheese) on the cabbage patty.
Position the second slice of bread on the sandwich buttered side up, cut in half if desired, and serve.