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.
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.
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 scallions.
You can use any brand of ramyun/ramen noodles, but I'm partial to Shin Ramyun; get the black label if you can.