To make the beef, toss it with all the marinade ingredients in a bowl or heavy duty zip-lock bag, coating the meat well. Refrigerate for a couple hours, or overnight.
To make the cheese sauce, whisk together the butter and flour in a medium saucepan over medium-high heat; whisk in the milk and bring to a simmer, stirring as it thickens. Stir in the cheese and kimchi juice, stirring until melted and smooth.
To cook the wontons, heat an inch or two of oil in a small, shallow pot or skillet until it’s hot but not smoking; cook about 4 pieces of the wonton at a time - or enough so that they aren’t overlapping - until they’re golden on both sides, turning with tongs. (This should only take a minute.) Transfer to a plate lined with paper towels and shower with salt while they’re still warm.
Set a skillet over medium-high heat, add a drizzle of oil and cook the bulgogi beef in batches, just until cooked through. (This will also take just a minute or two.)
Top the wonton chips with bulgogi beef, cheese sauce, chopped kimchi and fresh cilantro; add a squirt of gochujang if you like, and serve with lime wedges. Serves 4-6.