Whisk 1 tbsp gochujang, 2 tbsp mayonnaise, 1 tbsp rice vinegar, and 1 tsp sugar in a bowl. Add 2 cups cabbage and 1/2 cup carrot, toss to coat, and set aside.

Combine 1 lb ground chicken, remaining 1 tbsp gochujang, 2 tsp soy sauce, 1 tsp sesame oil, minced garlic, grated ginger, beaten egg, and 1/2 tsp salt. Mix gently, then shape into 4 patties with a slight indent in the center.

Heat a skillet over medium-high heat with vegetable oil. Cook patties about 5 minutes per side, until golden and 165°F (74°C) inside.

Toast buns in the same skillet, spread with any extra slaw dressing, add a patty, top with slaw, and serve right away.