Sauté the leeks in oil over medium heat until limp, 8-10 minutes.
Add the potatoes and stock, bring to a boil, lower heat, partially cover, and simmer, stirring occasionally, until potatoes are completely soft, about 30 minutes. If there isn’t enough liquid, add more stock, water, or milk,* ½ c at a time.
Stir in 2 c milk, salt, and pepper and simmer for 10 minutes.
You can serve the soup like this or put some or all of the soup in a blender or food processor for a smoother, heavier soup.
Garnish with additional pepper and parsley and serve.