Soak the rice cakes in cold water for 20 minutes. Skip this step if using fresh rice cakes that are already soft and don’t require cooking.
Mix all the sauce ingredients well in a large bowl. Add the chicken pieces and toss well to coat. Let the chicken marinate in this sauce while you prepare your vegetables.
Heat a large pan over medium high heat. Add a little oil, and then the sliced Korean sweet potato, as well as a little bit of the cabbage. The cabbage will release a little water, which will help the potatoes to steam a bit. Add a splash of water as well. Cover with a lid. Reduce the heat to medium low and let the potatoes and cabbage cook for about 2 minutes.
Add the rest of the cabbage. Make a well in the center. Raise the heat to medium-high. Add a little oil, along with the chicken and all the sauce. Cook for about 3-4 minutes, stirring occasionally to let the chicken start cooking. Incorporate the cabbage into the chicken. Lower the heat and add the rice cakes and cook until the chicken is cooked, and the potatoes and rice cakes are soft, but not mushy, another 3-5 minutes. You don’t want the potatoes to break down too much, or the sauce will get starchy. You can add a splash of water or chicken broth to the pan if it looks too dry.
Add the perilla leaves, scallions, cheese, and enoki mushrooms. Cover the pan, turn off the heat, and let stand for 1-2 minutes to allow the cheese to melt. Serve with extra perilla leaves or lettuce leaves for wrapping.