Place a medium skillet over medium-high heat and melt butter. Add sesame oil followed by the kimchi and sauté for 5 to 7 minutes until lightly caramelized. Remove from heat and allow to cool for at least 15 minutes.
Slather both sides of each slice of bread with butter.
Sprinkle some shredded gruyere onto the bottom half of a slice of bread and top with a heap of buttered kimchi. Top kimchi with another generous sprinkle of cheese and a small amount of green onions
Finish assembling the grilled cheese by topping with another slice of bread and gently pressing the entire grilled cheese together.
Place a cast iron skillet over medium heat and melt a couple tablespoons of butter. Once the butter has melted and the skillet is hot, carefully place the grilled cheese into the skillet and cook for 4 to 5 minutes or until the bottom is nicely toasted and some of the cheese has melted. Flip the grilled cheese and continue to cook for an additional 3 to 4 minutes or until the bottom is toasted and the remaining cheese has melted.
Transfer grilled cheese onto a cutting board. Allow the sandwich to sit for 3 to 5 minutes before cutting the half and serving.
