Soak: The day before making dashi, place kombu on a flat surface and use a damp paper towel to wipe away some of the salt. Pull the heads off the anchovies, open them up with your fingers, and use your thumbnail to scrape out the guts (which should be discarded). Place kombu, cleaned anchovies, and shiitakes in a large pot. Add the cold water. Cover pot and let soak at room temperature, 10-12 hours. (Alternatively, store in the refrigerator for up to a week before making dashi.)
Simmer: Place pot over high heat, uncovered, until the surface of the water begins to ripple, but do not allow it to come to a boil. Lower heat to a very low flame: the ideal temperature is a few degrees below a light boil (55 degrees C or 131 degrees F). Taste every 20 minutes to monitor the changes in flavor and texture; the dashi will usually be finished in around 80-90 minutes. You are looking for a sea essence, with a tiny hint of sweetness and a very silky texture (similar to Evian water), with no real seasoning. Be vigilant, as dashi will turn bitter if cooked too long. Strain and refrigerate in a covered container for up to 1 day. After 24 hours, the dashi will begin to lose some of its characteristic flavor, and will go slightly flat. Cooked and cooled dashi can be frozen for up to 2 weeks.