Make the kimchi slaw. In a bowl, combine cabbage, kimchi, carrot, and green onions. Stir in mayonnaise, rice vinegar, kimchi juice, and sugar if using. Season with a small pinch of salt.
Chill while you cook the burgers.
Mix the glaze. Whisk gochujang, honey, soy sauce, rice vinegar, sesame oil, and grated garlic until smooth. Set aside.
Form the patties. In a bowl, gently combine ground beef with salt, pepper, garlic powder, onion powder, soy sauce, and sesame oil. Don’t overmix.
Divide into 4 equal portions and shape into patties about 1/2 inch thick. Press a small dimple in the center of each to prevent doming.
Preheat the air fryer. Set to 375°F (190°C) for 3 minutes. Lightly oil the basket to prevent sticking.
Air fry the patties. Arrange patties in a single layer.
Cook for 5 minutes, then flip. Brush the tops with the gochujang glaze. Cook for another 3–4 minutes, depending on thickness.
Add cheese. In the last 1–2 minutes, place a slice of cheese on each patty.
Close the air fryer to melt.
Toast the buns. Split buns and air fry cut-side up for 1–2 minutes until lightly crisp. You can brush with a little butter for extra flavor.
Make a quick sauce. Mix mayonnaise with a little gochujang (about 1 tsp) for a creamy, spicy spread.
Assemble. Spread sauce on buns. Add the cheesy patty, spoon on a generous mound of kimchi slaw, and top with pickles and a sprinkle of sesame seeds if you like.
Close the burger and serve immediately.