Spread the softened butter evenly on one side of each slice of sourdough bread.
In a bowl, mix the chopped kimchi with the mayonnaise and sesame oil until well combined.
On the unbuttered side of one slice of bread, layer half of the shredded cheddar cheese.
Spread the kimchi mixture over the cheese, then top with the remaining cheddar cheese.
Place the second slice of bread on top, buttered side facing out.
Heat a skillet over medium heat and place the sandwich in the skillet. Cook for about 3-4 minutes until the bread is golden brown.
Carefully flip the sandwich and cook for another 3-4 minutes until the other side is golden and the cheese is melted.
Remove from the skillet, let cool for a minute, then slice in half and serve.