As I understand it, in Korea the standard operating procedure for this sandwich is to use soft white bread, a fair amount of sugar, and hefty doses of both ketchup and mayo. I take liberties with all those elements, hence “bastardized.” In fact, so long as cabbage and an allium are present, I often take yet more liberties, such as switching between onion and shallot (based on which is within easier reach), adding julienned red bell pepper and/or jalapeño (or substituting it for the carrot), and ditching the meat and/or cheese and the condiments. The moral of the story is that I consider the formula flexible and easy to bend to individual preference and ingredient availability.
Two ingredient tips, if I may: First, don’t use thin-sliced bread for this—it won’t hold up against the moisture in the vegetables. Second, if you opt for the cheese, go with a good melter like a young cheddar, Monterey or pepper jack, Gouda, or Fontina.
Once you’ve peeled the carrot, a quick and easy way to prep it is to simply keep using the peeler, rotating the carrot as you work, to create thin strips.
A wide, thin spatula (nonstick-friendly, if necessary) is the tool of choice for shaping and flipping the cabbage patty as it cooks.
All these options and notes might make this sound finicky, but it’s not. Make your choices and then make the sandwich to see how quickly and easily it goes.