Spread a thin layer of butter over one side of each slice of bread.
Turn 2 slices of bread over so that the buttered side is facing down. Divide the shredded cheese between these slices of bread.
In a small bowl, mix the chopped kimchi, scallions, gochujang, honey, and sesame oil.
In another bowl, mash the avocado. Mix in the lime juice, garlic powder, and salt.
Divide the kimchi over the slices of bread that are topped with cheese.
Heat a large skillet over medium heat. You may need to adjust the heat higher or lower depending on your stove. Carefully place the 2 prepared slices of bread with cheese and kimchi into the skillet. Cover the skillet and cook until the cheese is melted, about 3 to 4 minutes (see note 3).
Carefully divide the mashed avocado over the remaining slices of bread (see note 4). Don’t worry too much about having to spread the avocado all the way to the edges. The mash spreads out when you flip the sandwich over.
Top both sandwiches with the remaining slices of bread, avocado side down and buttered side up. Gently press down on the top slices of bread to adhere the bread to the rest of the sandwich. Using a large spatula, carefully flip both sandwiches over and cook the other side until the bread is toasted, another 2 to 3 minutes, uncovered. Transfer the sandwiches to a plate and serve immediately.