Bring a small pot of water to a boil over high heat. Drop the ramen noodles in and cook until the ramen noodles are soft, about 4-5 minutes only. You don't want to overcook the noodles.
Drain the ramen noodles and add the seasoning packet over the noodles. Toss to combine and set aside.
Butter one slice of bread on both sides and set aside. Using a cast iron skillet or non-stick skillet, heat one tablespoon of butter in your skillet over medium heat. Let the butter melt completely and then lay your slice of buttered bread into the skillet and top with a 1/2 cup of the shredded cheddar cheese. Cover the skillet with a lid and let the cheese melt until it's almost entirely melted, for about 2-3 minutes.
Next, lift the lid and top the melted cheese with half of the ramen noodles and about a 1/2 cup of the squeezed kimchi (or less depending on the size of your bread) and cover again with the lid for a minute. Butter both sides of the second slice of bread and place the bread slice on top of the kimchi, pressing down gently and flip your sandwich very carefully. You do not need to cover the pan again since your cheese is already melting nicely. You can flip one more time if desired (but not necessary) until you have nice, golden brown toasty bread on both sides.
Repeat everything for your second grilled cheese sandwich. Enjoy!