If the rice cakes are hard, soak them in warm water for 10-15 minutes until softened. In a large pan, combine water/broth, Gochujang, Gochugaru, sugar, soy sauce, and minced garlic. Stir well to dissolve the paste.
Place the pan on medium heat and bring the sauce to a simmer. Add the softened rice cakes. Stir constantly to prevent sticking and cook for 5-7 minutes until the rice cakes are chewy and the sauce thickens slightly.
Add the sliced onion, chopped cabbage, and sliced fish cakes to the pan. Continue to stir and cook for another 3-5 minutes, or until the vegetables are slightly softened.
Turn off the heat. Sprinkle the shredded mozzarella cheese over the Tteokbokki. Cover the pan with a lid for 1-2 minutes to allow the cheese to melt. Garnish with green onions before serving.
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