Soak the cylindrical rice cakes (tteok) in warm water for about 20 minutes to soften them.
In a large sauté pan, combine water, gochujang, coconut aminos, honey, minced garlic, and sesame oil. Bring the mixture to a simmer and stir.
Drain the soaked rice cakes and add them to the boiling sauce. Reduce the heat to medium and simmer until the rice cakes are soft and the sauce has thickened, about 10 to 15 minutes.
Once the rice cakes are soft and the sauce has thickened, stir in the shredded cabbage then top everything with shredded mozzarella cheese. (If you are adding hard boiled eggs, add them now).
Close the sauté pan with a lid to melt the cheese for about 1 to 2 minutes. Make sure to check under the lid to see if the cheese has melted to your liking.
Transfer the cheese tteokbokki to a serving dish and garnish with chopped green onions.