Clean the radish by scrubbing with a brush and/or scratching off the stubborn impurities with a knife. Cut the radish crosswise. Gather the slices and cut into about 3/4-inch strips.
Pour 5 cups of the broth to a bowl. Stir in one teaspoon of sugar and salt to taste. You can add 1 or 2 cups of dongchimi broth, if available, and reduce the beef broth by the same amount.
Cut the cucumber 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.
Bring a medium pot of water to a boil. Prepare an ice bath while water is boiling. Cook the noodles according to the package instructions. Drain and rinse again in icy cold water.
Place one serving of noodles in the middle of the bowl and top with the pickled radish, slices of beef, cucumber, and pear if using, and the egg half. Serve with vinegar and hot mustard paste.