In a large pot, bring the water and broth to a boil.
Add the rice cakes and fish cakes to the pot and cook for about 5 minutes until softened.
In a small bowl, mix together the gochujang, soy sauce, sugar, and garlic to make the sauce.
Add the sauce to the pot along with the cabbage, carrots, and green onions. Cook for another 5-7 minutes until the vegetables are tender.
Stir in the sesame oil and remove from heat.
Serve hot, garnished with toasted sesame seeds and sliced hard-boiled eggs if desired.