1. In a medium saucepan, combine the anchovies, kelp, and 5 cups (1200 ml) water and bring to a boil. Reduce the heat to maintain a simmer for about 20 minutes. Pass the anchovy stock through a fine-mesh strainer into a large, wide, deep skillet and discard the solids.
2. Return the stock to a simmer. Add the carrots, onion, garlic, chili paste, sugar, chili flakes and soy sauce, stir to combine and simmer for 5 minutes. Add the cabbage and noodles and simmer, stirring often, until the noodles are slightly softened, about 2 minutes.
3. Add the rice cakes and fish cake sheets and cook, stirring occasionally, until the noodles are just tender and the sauce has thickened slightly, about 5 minutes. Drizzle with the sesame oil, sprinkle with the sesame seeds and nestle the eggs in the sauce.
Serve immediately, topped with the spring onions.
You can use any brand of ramyun/ramen noodles, but I'm partial to Shin Ramyun; get the black label if you can.