To make the sweet and sour radish, clean the Korean Radish (1 pound) with a brush. Cut the radish into 2-3 inches long and 3/4 inch wide strips. You can use a mandoline to thinly slice, if available.
In a bowl, add the radish, Distilled White Vinegar (3 tablespoon), Granulated Sugar (3 tablespoon), and Salt (1 teaspoon). Mix well by hand until the sugar is dissolved. Taste and add more vinegar or sugar to taste.
Make the broth: In a large pot, bring the Beef Brisket (0.5 pound), Korean Radish (6 ounce), Onion (1/2), the white part of the Scallion (2), Garlic (6 clove), Fresh Ginger (3 slice), and Peppercorns (1/2 teaspoon) to a boil, uncovered, in 14 cups of water.
Reduce the heat to medium to medium low to keep it at medium boil, and skim off the scum. Continue to boil, covered, until the meat is tender, about 1 hour. Stir in Soy Sauce (2 tablespoon) when there are 10 minutes remaining.
Remove the meat and cool. Discard the vegetables. Cool the broth.
Pour 5 cups of the broth to a bowl (about 2-1/2 cups per serving). Stir in Granulated Sugar (1 tablespoon) and Salt (1 teaspoon). Keep it in the freezer for an hour or two until the broth becomes slushy. Keep the remaining broth in the fridge or freezer for later use.
Cut the Korean Cucumber (1/2) in half lengthwise. Thinly slice crosswise, lightly sprinkle with salt, and let it sit until the cucumber slices are wilted.
Thinly slice the beef against the grain. Thinly slice the Korean Pear (2 slice) into a half-moon shape if using.
Bring a medium pot of water to a boil. Prepare an ice bath while water is boiling. Cook the Naengmyeon Noodles (2 serving) according to the package instructions.
Drain quickly and shock in the ice water to stop cooking. Drain and rinse again in icy cold water until the noodles are very cold.
Make two one-serving size mounds, placing in a colander to drain.
Place one serving of noodles in the middle of the serving bowl and top with the pickled radish, slices of beef, cucumber, and pear if using, and the Egg (1).
Pour one half of the icy broth around the noodles. Repeat for another serving. Serve with Distilled White Vinegar (to taste) and Korean Hot Mustard Paste (to taste) on the side.