You will need:
An immersion blender or regular blender
4 medium leeks
6 medium yukon gold/yellow potatoes
3 tbsp. butter (vegan: 3 tbsp. olive oil)
1/2 tbsp. salt
1/2 tbsp. white pepper (or black pepper, white is better)
7-8 cups of vegetable stock
1/2 cup of buttermilk (vegan: 1 cup of butternut squash)
1 cup of heavy cream (vegan: 1 package of silken tofu)
2 medium carrots (optional)
1 large celery stalk (optional)
A 1" wide vertical slice of green bell pepper (optional)
A 1" wide vertical slice of red bell pepper (optional)
1/4 cup of chopped spinach or kale (optional)
Peel and small dice the potatoes, and set aside.
Cut the dark green parts and ends off of the leeks, and slice them vertically into thirds. Then, chop across the grain into 1/2" wide slices. Any other vegetables you're using besides the potatoes (including the squash for vegan vichyssoise), cut them into a fine dice.
In a large pot, add the butter (oil) and turn the heat to medium low. Add the cut-up vegetables, salt, and pepper. Keep this on low, stirring occasionally, for 25-30 minutes. Do not sautee/brown the leeks, because they won't taste right - you're trying to reduce the water like with cabbage, spinach, or greens.
After this has reduced a bit, add the potatoes and the cooking stock. Bring to a boil, stirring, then cover and turn the heat to medium-low for about 45-50 minutes, stirring twice.
Remove from heat, and let cool for 10 minutes. Add the buttermilk to the cream, then add to the soup (or add the silken tofu in dollops). Immersion blend until smooth (or use the blender, but mind the heat - additional cooling might be needed). Should have the consistency of and look somewhat like cake batter.
If making this vegan, bring the soup back up to a boil; otherwise, this can be served hot, but traditionally, vichyssoise is a cold soup. Chives go well on top, if you want. If you like meat, diced ham and bacon fit in well with this. Think of a soup that tastes kinda like Hungry Man potato pancakes or potato latkas, but with the consistency of ham and bean/pea soup.