In a skillet over medium heat, add 1 tablespoon of oil and the onion. Cook the onion for 4-5 minutes until soft and translucent, stirring occasionally. On the same skillet, add the remaining oil and heat the burgers for 2-3 minutes per side. Set aside.
In a bowl, combine mayonnaise, gochujang, tomato ketchup, and lemon. Mix well.
To serve, spread a little of the sauce on the bottom half of your choice of bun, top with onion, then the burger, and a good amount of kimchi and jalapeno slices. Finish with the top half of the bun.