Heat the heavy-bottomed pot over medium heat and add the butter, letting it melt until it starts to shimmer.
Add the sliced leeks to the pot and cook gently, stirring often, until they soften and turn a pale golden, about 8-10 minutes. The aroma of sweet, caramelized leeks will fill the kitchen.
Stir in the chopped potatoes, coating them with the leek butter mixture, and cook for another 2 minutes while stirring.
Pour in the vegetable stock, bringing the mixture to a gentle simmer. Cover the pot loosely and cook until the potatoes are tender when pierced with a fork, about 15-20 minutes.
Remove the pot from heat and use an immersion blender to puree the soup until smooth and velvety, or transfer in batches to a blender, blending carefully with vented lid.
Stir in the heavy cream, then season with salt and pepper to taste. Warm the soup gently over low heat, but do not boil to avoid curdling.
Let the soup rest for a few minutes to meld the flavors, then ladle into bowls and garnish with chopped fresh herbs if desired. Serve hot and enjoy the cozy, silky texture.