In a bowl, add the tuna (drain if there’s too much liquid), sesame oil, kewpie mayo, kimchi, and green onions. Mix until evenly combined.
In a separate bowl, mix together the melted butter, minced garlic, parsley, parmesan, black pepper, and salt.
Lay out the sourdough slices and brush one side of each slice generously with the garlic butter.
Flip the bread over so the buttered side is facing down. Sprinkle cheddar cheese onto the bread first, then spoon the tuna mixture on top, followed by another layer of cheddar cheese. Finish with the remaining bread slices, buttered side facing up.
Heat a skillet over medium heat. Cook the sandwiches for 3–4 minutes per side, covering loosely with a lid for part of the cooking time to help the cheese melt, until the bread is golden brown and the cheese is fully melted.
Remove from the pan, let rest for about 1 minute, then slice and serve hot.