Shred a cup of cabbage (about ¼ of a head usually) and slice a carrot into thin julienne strips. Then chop up half a small onion and a green onion or two (optional). Add all of these vegetables into a mixing bowl and given them a good stir. Then, add a teaspoon of salt, half a teaspoon of black pepper, and half a teaspoon of Korean gochugaru or red pepper flakes, and mix well. Finally, crack two eggs and mix well to combine.
Heat a large skillet and add half a tablespoon of butter. Then, add the vegetable-egg mixture and shape into a rough square or rectangular shape (roughly the size of your toast). Cook on either side for 2-3 minutes until the mixture turns golden brown and the shape holds. Then set aside. Note:You'll notice below that I use large rustic slices of bread for this sandwich. If you're using regular slices, I'd recommend using four slices instead of two and cutting the egg omelet into two square pieces for each sandwich.
In a small bowl, mix a teaspoon of gochujang with a tablespoon of mayonnaise to form a rich condiment. Set this aside.
Use half tablespoon of butter to slather both slices of toast on both sides. Add the rest of the butter to the skillet. Place the two slices of toast and toast gently until golden brown on both sides, about two minutes, or slightly less than your desired level!
Remove the toasts from the heat and place on a plate. Add a slice of cheddar (or shredded cheddar) on one slice and the Korean egg omelet on the other. Slather the gochujang-mayonnaise on the omelet side and close the sandwich. Place this back on the skillet and toast about 30 seconds on either side to melt the cheese a bit.
Slice into two triangular halves if you're using a large rustic slice of bread. Your Korean street toast is ready to go!
Bread: I like using soft but sturdy breads as the best vehicle for this sandwich. However, sourdough or brioche, or really any type of bread works! You'll notice that the slices I've used are fairly rustic and almost as big as my palm. If you're using regular slices, I'd recommend slicing up the omelet into two pieces to make two separate sandwiches (versus slicing one sandwich into halves).
Veganizing: To make a vegan version of this recipe, use vegan butter and mayonnaise, and replace the egg with Just Egg or a similar egg replacement!
Number of eggs: A lot of recipes call for just one egg in this recipe, but I prefer using two since I find that it actually makes for a more filling breakfast!