Heat a cast-iron skillet over medium heat. Top 1 piece of bread with 1/4 cup grated Cheddar, kimchi, the remaining 1/4 cup grated Cheddar, and then the other slice of bread. Spread 1 tablespoon mayonnaise over the top slice, then add to the skillet mayonnaise-side down and cook for 3 minutes, pressing slightly to flatten. Spread the remaining 1 tablespoon mayonnaise on the top slice of bread, flip, and cook until golden, another 3 minutes. Quarter the sandwich and serve.