Step 1In a large stock pot, bring 10 cups of water, the beef brisket, garlic, ginger, green onions, serrano chile, 3 tablespoons of the distilled vinegar and the soy sauce to a boil. Once it comes to a boil, reduce the heat and simmer for 35 minutes uncovered.
Step 2Remove the meat using tongs and set it aside on a cutting board to dry, wrapped in paper towels. Place something heavy, such as a cast-iron skillet, on top of the meat to keep it flat. When it's cool enough to handle, quarter the brisket. Slice a quarter of the brisket into one-eighth-inch-thick slices and reserve the rest for another use.
Step 3Skim any scum off the top of the broth. Remove the broth from the heat and strain into a large bowl. To the broth, add the dongchimi liquid, the remaining 2 tablespoons distilled vinegar, 2 teaspoons salt and 2 1/2 teaspoons sugar. Stir to thoroughly combine and chill in an ice water bath, or refrigerated for about 2 hours, until very cold.
Step 4Prepare a large pot of boiling water and cook the noodles according to the package directions (about 2 minutes) until the noodles are just al dente, flexible but springy in texture and chewy. Drain immediately and rinse with cold water to stop cooking.
Step 5Divide the noodles into four large soup bowls. Pour the chilled broth over each pile of noodles. Top each bowl evenly to taste with the beef, pear, daikon, cucumber and half a hard-boiled egg. Serve accompanied by bowls of vinegar, hot Asian mustard and sugar to garnish.