Strain the kimchi juice through a sieve, then mix it with the rest of the broth ingredients (except the ice). Put the broth in the freezer while you prepare the other ingredients.
Boil the egg for the garnish and cut it in half, and thinly julienne the cucumber.
Mix the kimchi with the sugar and sesame oil to make the kimchi topping.
Boil the somyeon noodles for the time indicated on the package (about 3 minutes). Adding 1 cup of cold water when the water comes to a boil makes the noodles chewier.
Scrub the boiled noodles thoroughly under cold water to remove the starch completely, then drain.
Put the noodles in a bowl and top with the seasoned kimchi, cucumber, and boiled egg.
Pour the cold broth over the noodles, add the ice, and finish with a sprinkle of toasted sesame seeds.