Place the rice cakes in hot water and soak for 15 to 20 minutes while preparing the anchovy stock.
Add the dried anchovies, dried kelp, fish cake sheets, and water to a large skillet or saucepan.
Bring to a boil over high heat, then reduce to a gentle simmer and cook for 15 minutes.
Remove the fish cakes and set them aside to cool slightly.
Strain the stock, discarding the anchovies and kelp.
Measure 1 cup of the stock and return it to the skillet. Reserve the remaining stock for another recipe.
Heat the stock over medium heat and whisk in the gochujang, gochugaru, and sugar until smooth.
Drain the soaked rice cakes and add them to the sauce.
Cook for about 5 minutes, stirring frequently, until the rice cakes become tender and the sauce thickens.
Slice the fish cakes into bite-sized strips and add them to the skillet.
Gently toss everything together for another minute.
Serve immediately topped with hard-boiled eggs, chopped scallions, and toasted sesame seeds if desired.