Cut the chicken into bite-size pieces. Roughly chop the cabbage (and onions, if using).
In a bowl, combine gochujang, gochugaru, soy sauce, honey, mirin, and garlic. Mix until smooth.
Add the chicken, cabbage, and onions to the sauce and toss until evenly coated.
Heat a large pan over medium-high heat and drizzle with oil.
Add the marinated chicken and vegetables. Let the chicken sit undisturbed until it gets color. Cook, stirring occasionally for 8-10 minutes or until the chicken is fully cooked and the sauce has thickened and caramelized slightly.
Turn off the heat and generously top with mozzarella cheese.
Cover with a lid and let the cheese melt, or broil briefly until bubbly and lightly browned.
Garnish with sesame seeds.