Combine gochugaru, gochujang, brown sugar, ginger, garlic, soy sauce, and black pepper in a medium bowl and mix well. Add the chicken and stir until well coated
If using the rice cakes, swirl the oil into a large, oven-safe skillet set over medium-high heat. Wait for the oil to shimmer. Add the rice cakes and cook, turning often, until they are a little crisp, about 3-4 minutes. Transfer to a small bowl and set aside.
Add the chicken mixture to the pan (do this once oil shimmers if not using rice cakes) along with 1/4 cup water. Cover and cook over medium-high heat until chicken is cooked through, 8-10 minutes, stirring occasionally. Add the rice cakes halfway through if using. Heat the broiler in your oven
Remove chicken from heat and cover pan with mozzarella. Slide pan under the broiler. Cook until the cheese is melted and browned in spots, about 2-3 minutes. Serve immediately, with scallions as garnish