
Heat a heavy skillet over medium-low. Thinly spread 1 side of each slice of bread with 1/2 tablespoon mayonnaise. Place the bread, mayonnaise side down, in the skillet and divide the mozzarella evenly over the slices.
When the cheese has just melted (6 to 10 minutes), add the kimchi to one side. Use a spatula to top with the other slice of bread, cheese side down. Press to meld, and let cook, covered, flipping as needed to prevent burning, until the bread is crusty-brown and sandwich is warmed through (2 to 4 minutes).