If your rice cakes are frozen, soak them in warm water for 15 to 20 minutes while you prep everything else, then drain. Fresh rice cakes can go straight into the pan.
Combine the water or stock, gochujang, gochugaru, sugar, soy sauce, and garlic in a wide, shallow pan. Stir until the gochujang dissolves, then bring to a boil over medium-high heat.
Add the rice cakes and onion. Reduce the heat to medium and simmer, stirring often so the rice cakes do not stick, for 8 to 10 minutes. The sauce will thicken and the rice cakes will soften.
Stir in the fish cakes and simmer 5 to 7 minutes more, until the sauce is glossy and thick enough to coat a spoon and the rice cakes are tender all the way through. Adjust sugar, gochugaru, or soy sauce to taste.
Fold in the scallions and the halved eggs, if using, and simmer for 1 minute more. Drizzle the sesame oil over the top, scatter with sesame seeds, and serve straight from the pan while bubbling.