Put some oil into the cast-iron pan and put it on the stove over medium heat.
Spread the sweet potatoes and rice cakes in the pan.
Put the sliced cabbage and onions on top, then the marinated chicken.
Let it simmer for a bit, then give it a good stir to distribute the marinade. Keep it on the stove over medium heat, stirring occasionally.
Note: This step might take a while (20+ minutes) if you're making a larger serving amount. The pan is very overcrowded, which is completely fine. The cabbage releases a lot of moisture – keep going until it's less of a soup and more of a sauce. Importantly, the chicken must be done, the rice cakes should be chewy, and the sweet potato should have no more crunch.
Optional: Preheat the oven to ~180 degrees to help melt the cheese later.
When the consistency looks good, take it off the heat and stir in all the instant noodles.
Part it in the middle and fill with cheese. If you want it to melt nicely, put the pan in the pre-heated oven for just 3-5 minutes.
Serve immediately and sprinkle with green onions. (If you have a stove to keep it warm on the table where it is eaten, that would be perfect.)