Tteokbokki: Spicy Korean Rice Cakes

To make homemade anchovy broth: In a large pot over medium-low heat, toast anchovies until very lightly golden, stirring frequently, 2 minutes. Add in water, mushrooms, and salt and bring to a boil over high heat. Reduce heat to medium-low and continue to simmer for 15 minutes. Remove from heat, then add in kelp and let steep for 10 minutes. Strain, reserving kelp, mushrooms, and fish for another use, if desired. Meanwhile, if using hard, refrigerated rice cakes: Separate cakes if they are stuck to each other, then rinse in cool water and drain. Transfer to a large bowl and cover cakes with 3 cups of hot water. Let soak for 15 minutes, until softened. Drain and set aside. In a large pot over medium heat, heat oil. Add gochugaru, gochujang, garlic, soy sauce, vinegar, and honey and stir until combined and fragrant, 1 minute. Add 2 cups broth and bring to a boil. Add in soaked rice cakes and cook until cakes are tender and soft, stirring often, about 10 minutes.  Add in remaining broth as needed to loosen sauce to desired consistency. Add in fish cakes, green onions, and sesame oil, and cook until onions are softened, about 5 minutes. Garnish with more green onions and sesame seeds before serving.