The combination of grilled beef in a sweet and salty marinade and kimchi is about as classic as it gets. Think: kalbi or bulgogi. Even the combination of marinated beef, kimchi, and cheese is not unprecedented. Just take a look at the Korean taco craze. Koreans will take slices of American cheese and melt them into their kimchi jjigae. I had an awesome bulgogi hoagie at Broadway Cafe in Ann Arbor last year. Even the Serious Eats book has a recipe for kimchi quesadillas.
So to put kimchi on a burger seems only natural. Once upon a hangover, I made myself a burger that combined not just kimchi, but pickled jalapeños, cheese, and eggs fried in duck fat. That monstrosity managed to allay my headache for a few hours, but it certainly broke most if not all of my Principles of Burger Topping.
With a clear mind, we can do better.
For starters, chopped kimchi and American cheese are a must for any kim-cheeseburger. After that? I decided to go with a quick pickled sweet-and-salty slaw of carrots and cucumbers. They add a nice crunch (which my go-to super-fermented kimchi lacks), and a sweetness that pairs nicely with the heat and garlic of the kimchi. I make it the way I'd make quick pickles for a Vietnamese bành mí: rub some sugar and salt into vegetables cut into matchsticks (disks work just fine), add a touch of vinegar, and let it sit for half an hour or so—just about the time it takes to light up the grill and cook off my burgers.
For the patties themselves, thick and beefy is the way to go. I generally like to grind my own beef blend, but for a burger with many toppings, fresh ground chuck works fine. (Just make sure it's got enough fat!)
And for that sweet glaze? Sweet kalbi-style marinade made with soy sauce, sugar, sesame oil, and plenty of garlic and ginger is the way to go. (You can also use store-bought bulgogi or kalbi marinade if you'd like.) Painting it onto the burger at the start of grilling leads to acrid burnt flavors. Instead, I find that applying it in a few coats just for the last few moments gives you a beautiful lacquer-like finish with just a hint of caramelization.