Step 1 Put the cut potatoes in a medium saucepan with the salt and cilantro stems, cover with about 4 cups cold water or more to cover by a half inch or so. Bring to a boil, then boil until the potatoes are tender, about 12 to 15 minutes. Discard the cilantro stems; reserve the potatoes and their cooking water.
Step 2 While the potatoes cook, mince the cilantro, anchovies and garlic together on a cutting board until very fine, almost like a paste. Reserve.
Step 3In small saucepan, saute the sliced leeks in the butter until tender but not browned, about 5 minutes.
Step 4In a food processor or blender, combine the leeks, potatoes and cooking water and pulse to puree, in batches, until very smooth. Do not overprocess the mixture or the potatoes will be gummy. Strain through a sieve, pressing on the solids.
Step 5Return the potato puree to the saucepan, add the cream and bring to a simmer. Taste for salt, remembering that the anchoiade can be salty; if the soup is too thick, add more cream or water.
Step 6Serve in soup bowls and divide the cilantro-anchovy chop among the servings, a spoonful swirled on top of each.