Separate the rice cakes into individual pieces.
In a large pot or saute pan, bring water and dried mushrooms to a boil over medium-high heat. [Maanchi's version calls for dried anchovies rather than mushrooms.] Once the mushrooms have softened, add the rice cakes, fish cakes, gochujang, and sugar to the broth. Simmer for approximately 25 to 35 minutes, stirring constantly.
When the rice cakes are al dente and the sauce has slightly thickened, add the green onions, white onions, and zucchini. Season the broth with additional sugar and/or gochujang to taste. If using ramen noodles, add to the mixture after the vegetables are fully cooked. [Dukboki with ramen added is called rabukkee.] Finally, stir in the hard-boiled eggs.
The dukboki is ready when the sauce is thick enough to coat the rice cakes, and the rice cakes are plump and soft all the way through. Sprinkle with sesame seeds and serve immediately.
Makes 6 to 8 servings.