Leeks are notorious for holding dirt in their upper parts. For this recipe, where everything will end up being pureed, I recommend this method: Cut each leek in two, cutting where the first leaves begin to separate from the core. Pull apart the leaves and rinse them under running water to remove most of the dirt. Thinly slice the leek greens across their grain, Cut the white part of the leek lengthwise and then slice each half thinly across the grain. Put all of the leeks in a large bowl of water and stir, so that any grit sinks to the bottom of the bowl. Transfer the leeks from the bowl to the cooking pot before dumping the water.
Add a couple of tablespoons of olive oil and a teaspoon of salt to the pot, and begin to sauté over medium-high heat. Stir fairly often to bring softened leeks to the top and expose the raw leeks to the heat of the bottom of the pan. Continue until the leeks are softened, about 10-12 minutes.
If the potatoes have a yellow or light colored skin, peeling them is optional. Clean the potatoes and cut into 3/4" chunks.
Add the potatoes to the pot and add enough stock to just reach the top of the other ingredients. Turn the heat to high and bring to a boil. Lower the heat to low, cover the pot, and simmer.
After about 10 minutes, check on the status of the pot. If there appears to be excess liquid, let the simmering continue without the cover. If most of the ingredients are above the stock line, add more stock. Once the liquid level looks right, cover and continue simmering.
Continue to check every 10 minutes, adjusting the liquid levels, simmer for at least 30 minutes total.
Once the ingredients are fully cooked, use an immersion blender to puree everything into a smooth soup. Pureeing leeks is more of a challenge for an immersion blender than with other cream of soups, so be patient and take your time. You may also use a traditional blender, but read the notes below.
Final adjustments are made at this time. If the soup seems too thick, add stock or water. Taste for seasoning and creaminess. For a richer mouthfeel, add cream at this point. For brightness, you can add a teaspoon or two of acid such as lemon juice or vinegar. Add salt and pepper and other seasonings to taste.