Simmer the anchovies and kelp in the water for 12 to 15 minutes over medium heat, removing the kelp after 5 minutes so it doesn't turn bitter.
Strain out the anchovies. Season the clear stock with soup soy sauce and minced garlic, then keep it warm over low heat.
Separate the egg. Fry the white in a lightly oiled pan over low heat until just set, in one thin sheet, then do the same with the yolk in a separate thin sheet. Let both cool slightly, then slice into fine strips.
Sauté the julienned zucchini in the same pan with a pinch of salt for 1 to 2 minutes, just until it softens slightly. Set aside.
Bring a separate pot of water to a boil and cook the somyeon for 2 to 3 minutes, until just tender. Somyeon overcooks fast, so start checking early.
Drain the noodles and rinse immediately under cold running water, rubbing the strands gently between your hands. This step removes surface starch and stops the noodles from turning gummy or over-softening in the hot broth.
Divide the rinsed noodles between two bowls. Ladle the hot broth over top.
Garnish with the julienned egg, sautéed zucchini, and gim. Finish with a few drops of sesame oil and serve right away.