Use half a tablespoon of butter to slather both slices of toast.
Add the rest of the butter to the skillet.
Place the two slices of toast, in the skillet, and toast gently until golden brown on both sides; about two minutes, or slightly less to meet the desired level of doneness.
Remove the toast from the heat and place on a plate.
Add a slice of 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 both sides, to melt the cheese a bit.
Remove from skillet and slice into two triangular halves and serve.