In a large pot, combine the water, dried kelp, and dried anchovies. Bring to a boil, then reduce the heat and let it simmer for about 10 minutes.
Remove the kelp and anchovies from the pot and discard them, leaving the broth.
Add the rice cakes to the broth and simmer for about 5-7 minutes, or until they start to soften.
Stir in the gochujang, gochugaru, soy sauce, sugar, and minced garlic. Mix well until the sauce is evenly distributed.
Add the thinly sliced onion, chopped green onion, and fish cakes to the pot.
Continue to simmer for another 5-7 minutes, or until the sauce thickens and the rice cakes are fully cooked and chewy.
If using, add the hard-boiled egg to the pot and let it heat through for a few minutes.
Stir in the sesame oil and mix well.
Garnish with sesame seeds.
Enjoy your meal!