Make the broth at least 2 hours ahead, ideally the night before: stir together the beef broth, dongchimi juice, rice vinegar, sugar, and salt, then chill until properly cold, not just cool from the fridge.
About 45 minutes before serving, taste the broth and adjust the seasoning. It should taste slightly too salty and sour on its own — the noodles are bland and will pull that seasoning back out.
Bring a large pot of water to a rolling boil and cook the naengmyeon noodles according to the package, usually 3 to 4 minutes.
Drain the noodles straight into a bowl of ice water and rub them gently between your hands under the cold water for about 30 seconds. This is what gives the noodles their snap.
Divide the noodles between two well-chilled bowls, twisting each portion into a tight coil in the center.
Pour the icy broth around the noodles, filling each bowl about three-quarters full.
Top with the cucumber, pear, radish kimchi, and half an egg per bowl, then drop in a handful of ice cubes or crushed ice.
Serve immediately with kitchen scissors for cutting the noodles at the table, plus mustard and extra vinegar on the side.