To begin, soak the rice cakes in warm water for 10 minutes to soften them. This helps ensure they cook evenly and absorb the flavours of the sauce. Meanwhile, prepare your other ingredients by slicing the fish cakes, onion, and green onions, and set them aside.
If you want to use anchovy broth for enhanced flavours, prepare it now. Simmer 2–3 dried anchovies and a small piece of kombu (dried kelp) in 3 cups of water for 10 minutes. Strain the broth and keep it warm. This adds depth to the dish, but plain water works as well.
In a medium sized pot or deep skillet, heat the broth or water over medium heat. Add gochujang, gochugaru, soy sauce, sugar, sesame oil, and minced garlic. Stir well to dissolve the gochujang and create a smooth sauce. Bring the mixture to a gentle simmer.
Once the sauce begins to bubble, add the soaked rice cakes, fish cakes, onion, and carrot (if using). Stir gently to coat all the ingredients in the sauce. Reduce the heat to medium-low and simmer for 10–15 minutes, stirring occasionally to prevent sticking.
Check the rice cakes for doneness by piercing them with a fork; they should be soft and chewy. If the sauce reduces too quickly, add a little more water or broth to maintain a slightly thick consistency. Adjust the seasoning with more soy sauce or sugar, if desired.
Add the hard-boiled eggs and half of the chopped green onions. Stir gently to incorporate them into the dish. Allow everything to simmer for another 2–3 minutes so the eggs can soak up some of the sauce.
Remove the pot from heat and let the tteokbokki rest for a minute to thicken slightly. This ensures the sauce clings to the rice cakes and fish cakes, enhancing each bite's flavours.
Serve the tteokbokki hot in a large communal dish or individual bowls. Garnish with the remaining green onions and sprinkle toasted sesame seeds on top. Pair with a side of kimchi or a refreshing cucumber salad. For a traditional experience, enjoy with a chilled barley tea or soju.