If using frozen rice cakes, soak them in warm water for 20-30 minutes until softened. Drain and set aside.
In a bowl, combine gochujang, gochugaru, soy sauce, sugar, and minced garlic. Mix well to form a smooth sauce.
In a large skillet or pan, bring the dashi or soup stock to a boil over medium-high heat. Add the sliced onion and cook until slightly translucent, about 2 minutes.
Add the softened rice cakes and fish cakes (if using) to the pan. Pour in the prepared sauce and stir to combine. Reduce heat to medium and let it simmer, stirring occasionally, until the sauce thickens and the rice cakes are tender, about 8-10 minutes.
Once the sauce has thickened, reduce the heat to low. Sprinkle the shredded mozzarella cheese evenly over the top. Cover the pan with a lid and let it cook until the cheese melts, about 2-3 minutes.
Drizzle sesame oil over the top. Garnish with toasted sesame seeds and chopped green onions. Add halved hard-boiled eggs if desired.