Mix the flour and salt, then add the water gradually, stirring with chopsticks or a fork until it comes together into a shaggy mass.
Knead 8 to 10 minutes until the dough is smooth and springs back slowly when you press it. Rub the surface with the oil, wrap tightly, and rest it at room temperature for at least 30 minutes.
Simmer the anchovies and kelp in the 1.4 L water for 15 minutes, then strain and discard the solids.
Roll the rested dough out thin, about 2 mm / 1/16 in, dusting with flour as you go so it doesn't stick to itself.
Fold the sheet loosely into thirds and slice crosswise into noodles about 4 mm / 1/6 in wide. Shake them apart with a little extra flour so they separate instead of clumping.
Bring the strained stock back to a boil, add the garlic and potato, and simmer 5 minutes until the potato is nearly cooked through.
Shake the excess flour off the noodles and drop them straight into the boiling stock — that flour coating is exactly what gives kalguksu broth its characteristic light cloudiness, so don't rinse it off first.
Add the zucchini and cook 4 to 5 minutes, stirring occasionally so the noodles don't settle and stick to the bottom of the pot.
Season with soy sauce and salt to taste, finish with sesame oil, and top with scallions.