Instructions

1. ring a large pot of water to a boil. Add the rice vermicelli and cook, stirring occasionally, until tender but firm to the bite, 1–3 minutes. Drain in a strainer and rinse under cold water. Set aside.
2. Rinse crabs thoroughly under cold running water. In a 10-quart pot, bring 6 quarts of water to a boil over high heat. Add crabs and cook, covered, until the shells turn a vibrant reddish orange, about 8 minutes. Using tongs, transfer crabs to a large bowl; let cool slightly. Set the pot of crab broth aside.
3. Working with 1 crab at a time, use a table knife to pry off top shell. With a small spoon, scoop out the brownish-yellow innards (also known as the mustard) and roe (if present) attached to the inside of top shell; transfer to a bowl. Place top shell in pot of reserved crab broth. Pull off and discard fingerlike gills attached to bottom of crab. (Rinse out any sand lodged behind gills.) Break the bottom shell in half, extract the meat with your fingers, and add it to the bowl of innards. Transfer bottom shell to the pot of reserved broth. Crack claws; add to broth. In the end, you should have about 1 cup of the crab mixture; set aside.
4. Put the pot of crab broth and shells over high heat; bring to a boil. Lower the heat to medium and simmer, without stirring, about 30 minutes. Strain the broth through a fine-mesh sieve. Transfer 15 cups broth to a 6-quart pot; set aside. (Reserve remaining broth for another use.)
5. Meanwhile, combine the dried shrimp with 1 1⁄2 cups water; bring to a boil. Lower the heat to medium-low; simmer until soft, about 15 minutes. Strain over a bowl, reserving shrimp and liquid separately.
6. Purée the softened dried shrimp in a food processor to a coarse paste. Add reserved crab mixture; pulse to combine. Transfer to a bowl and stir in the egg whites. Set crab mixture aside.
7. In a small saucepan, heat 3 tbsp. oil over medium heat. Add green scallion pieces; cook until slightly wilted, about 3 minutes. Transfer to a plate. Return pan to medium-low heat; heat remaining oil. Add paprika; cook, stirring frequently, until it turns a deep, brick red color, about 5 minutes. Set pan aside.
8. Combine crab broth and shrimp liquid; bring to a boil. Reduce heat to medium-low; add shrimp sauce, sugar, and tomatoes and simmer the soup until tomatoes are slightly softened, about 20 minutes.
9. Break reserved crab mixture into little pieces. Sprinkle over soup; they'll drop to the bottom. Simmer soup, without stirring, until mixture floats to surface to form somewhat puffy top layer resembling cooked egg whites, 3–5 minutes. Drizzle paprika mixture over soup; sprinkle with cooked and raw scallions.
10. To serve: Mix the sprouts, mint, and perilla in a bowl; set on the table along with the shrimp and chile sauces. Divide the reserved noodles between serving bowls and ladle some of the soup into each bowl. Garnish bowls with the herb mixture and sauces to taste.