Peel the potatoes and puree them briefly, in batches, in a food processor or blender, making sure that they aren't liquefied. Add to a heavy-bottomed pot.
Trim the leeks, cutting off the rough green parts of the stalks. Immerse the trimmed leeks in cold filtered water, add the kosher or sea salt, and let stand for a half hour. Rinse with cold filtered water, removing any impurities. Separate the leaves, chop into 2-inch length rectangles, and add to the pot.
Peel the parsnip, parsley root, and carrots. Continuing to use the peeler, shave the parsnip, parsley root, and carrots, depositing the shavings into the pot.
Fill the pot with cold filtered water, to the level just barely covering the ingredients. Bring the pot to a boil, then cover, and lower the heat to a very low flame. Simmer the soup for 2 hours. Turn off the heat and it's ready to serve.