Make the broth. Add water, anchovies, kelp, onion, and garlic to a pot.
Bring to a gentle boil, then simmer 15 minutes. Remove kelp at 10 minutes to avoid bitterness. Strain and return the clear broth to the pot.
Season with salt and soy sauce.
Mix the dough. In a bowl, combine flour and salt. Pour in hot water and stir with chopsticks or a fork until shaggy. Add oil if using.
Knead 5-7 minutes until smooth and elastic, adding a sprinkle of water if dry.
Rest the dough. Wrap or cover and let it rest 20-30 minutes. This relaxes gluten and makes rolling easier.
Prep vegetables. Julienne the potato and zucchini. Slice the cabbage and green onions.
Keep them ready by the stove.
Roll and fold. Dust your surface and rolling pin. Roll the dough into a large, thin sheet, about 1/16-1/8 inch thick. Dust lightly with flour, then fold the sheet into thirds like a letter.
Slice the noodles. Using a sharp knife, cut the folded dough into 1/4-inch wide strips.
Unfurl and toss with a little flour to prevent sticking. Aim for rustic, not perfect.
Cook the veggies. Bring the broth back to a boil. Add potato and cabbage, simmer 3-4 minutes until starting to soften.
Cook the noodles. Shake off excess flour and add noodles to the pot, gently stirring to separate.
Simmer 3-4 minutes until tender but springy.
Finish the soup. Stir in zucchini and green onions. Cook 1-2 minutes more. Add sesame oil and a few grinds of black pepper.
Taste and adjust salt or soy.
Serve hot. Ladle into bowls. Top with torn gim and extra scallions if you like. Enjoy with kimchi on the side.