Whisk the mayonnaise, ketchup, dill relish, Sriracha, horseradish, rice vinegar, salt, and pepper in a small bowl until smooth. Refrigerate while you prep the rest.
Place the kimchi in a sieve and press firmly to remove excess liquid, then chop and toss with the sesame oil. Pat dry with a paper towel to keep the sandwiches crisp.
Warm the corned beef: pile it into a dry skillet with 1 tablespoon water, cover, and heat over medium for 1–2 minutes until steamy. Keep covered off heat.
Butter one side of each bread slice. Spread a thin layer of the chilled sauce on the unbuttered sides.
Preheat a large nonstick or cast-iron skillet over medium-low heat. Set two slices of bread in the pan, buttered sides down and sauced sides up. Top each with 2 slices Swiss. Cover and cook 2 minutes to start melting.
Divide the warm corned beef over the cheese, spoon the kimchi on top, and drizzle with a little more sauce to taste. Cap with the remaining bread, buttered sides up.
Grill the sandwiches 3–4 minutes per side over medium-low, pressing gently with a spatula after the first flip, until the bread is deeply golden and the cheese is fully melted. Rest 1 minute, then slice and serve.