In a saucepan, melt half the butter over low heat. Add the leeks and cook, stirring occasionally, until softened, about 5 minutes. Add the potatoes and cook for 3 minutes.
Increase the heat to medium, pour in the stock and bring to boil. Reduce to a simmer, cover, and cook until the leeks and potatoes are tender, 35 to 40 minutes. Remove the pan from the heat and add the remaining butter in small pieces, stirring until it’s all incorporated.
Season with sea salt and pepper to taste. Transfer to a blender and puree until smooth. Serve in warm bowls.