In a large non-stick skillet, melt one-third of butter over medium heat until foaming subsides. Add both bread slices and cook, swirling occasionally, until pale golden brown on bottom side, about 2 minutes.
Transfer to a work surface toasted side up. Place one cheese slice on top of one slice of bread. Top with kimchi and second cheese slice. Close sandwich, with both toasted sides facing inwards.
Melt one more piece of butter in the skillet and reduce heat to medium low. Add sandwich and cook, swirling occasionally, until deep, even golden brown, about 5 minutes. Remove sandwich using a flexible metal spatula. Add the remaining butter. Return sandwich to skillet toasted-side up. Season with salt. Cook, swirling occasionally, until second side is deep, even golden brown and cheese is thoroughly melted, about 5 minutes. Serve immediately.