Heat a medium sauté pan oven medium heat and add butter. Once butter is melted, sauté kimchi in the butter until lightly browned and heated through. Set aside.
Place a sieve over a large bowl. Carefully drain water from potatoes through sieve reserving the water and squeezing moisture out of the potatoes. Let the white starch settle on the bottom of the water and then pour off water, reserving the starch. Place potato mixture back into a bowl, add onions and use paper towels to ring out as much moisture as possible. This will ensure a crispy latke. Then add starch back in and mix in garlic, egg, flour, chives, sesame seeds and salt. Mix to combine.
Heat the sauté pan back up and add oil. Once oil is hot (latke batter should sizzle when it hits the oil) add in about half of the latke mixture and flatten to a round pancake. Fry until golden brown, about 3 minutes. Flip and fry the other side. Let drain on paper towels.
Repeat with remaining batter. Fry one side, then flip and top with cheese, kimchi and other latke. Fry until latke is browned on the bottom and cheese is melted. Cut in half and serve hot!