In a bowl, whisk together the soy sauce, honey, brown sugar, sesame oil, garlic, ginger, and gochujang.
Add the steak to a sealable bag and pour the marinade on top. Massage the marinade into the steak to coat, then marinate it for at least 30 minutes or up to 8 hours.
Heat 2 tablespoons vegetable oil in a cast iron skillet over medium heat.
Add the steak, working in batches if needed, and sear it until deeply browned. Remove it from the skillet and reserve.
To build the grilled cheese sandwich, spread butter on one side of each slice of bread.
Add 2 slices of gouda and ½ cup cheddar to the other slice of bread.
Add ½ cup of shaved steak.
Add ¼ cup kimchi and a sprinkling of scallions.
Top with another slice of bread.
Heat the remaining oil in a skillet over medium heat.
Add the assembled sandwich and toast until browned, about 2 minutes.
Flip and toast the other side. Repeat with remaining sandwiches.