Heat a large pot over medium heat, add 3 tablespoons unsalted butter, and let it melt until foaming, about 2 minutes.

Add sliced leeks and cook, stirring frequently, for 5 to 7 minutes until soft and translucent.

Add diced potatoes and 4 cups broth to the pot, bring to a boil over medium-high heat, then reduce to medium and simmer for 12 to 15 minutes until potatoes are tender.

Remove the pot from heat and let the mixture cool for 2 minutes before blending.

Transfer soup to a blender in batches, filling no more than halfway, and blend each batch for 60 to 90 seconds until smooth; alternatively use an immersion blender until no lumps remain.

Strain the blended soup through a fine-mesh strainer into the pot, pressing gently to extract a silky puree.

Stir in 1 cup whole milk or half-and-half, then season with 1/2 teaspoon fine sea salt and 1/4 teaspoon white pepper, adjusting to taste.

Transfer soup to a container, cover, and refrigerate for at least 2 hours until fully chilled.

Garnish with chopped chives or chervil and add a dollop of crème fraiche or sour cream before serving, if desired.