Drain and chop the kimchi. Squeeze out excess liquid and set aside.
Whisk mayonnaise, ketchup, gochujang, relish, and rice vinegar in a small bowl.
Spread butter on one side of each bread slice.
Heat a skillet over medium. Place two bread slices, buttered side down, in the pan.
Layer cheese, corned beef, and kimchi on the bread. Spoon on some dressing.
Top with more cheese and the remaining bread, buttered side up.
Griddle until golden and melty, 3–4 minutes per side. Adjust heat as needed.
Rest 1 minute. Slice and serve with extra dressing for dipping.