Using a sharp knife, trim off the dark green tops of the leeks, then slice the white and light green parts into thin rounds. Rinse thoroughly in cold water to remove any grit, then drain well.
Peel the potatoes and cut them into small, even cubes about 1 cm (½ inch). This helps them cook quickly and evenly.
Place a large heavy-bottomed pot over medium heat and add a tablespoon of butter. Once melted and bubbling softly, add the sliced leeks. Sauté gently for about 8-10 minutes, stirring occasionally, until they become translucent and fragrant, filling your kitchen with a sweet, mild aroma.
Add the cubed potatoes to the pot and stir to coat them with the leek mixture. Cook for another 2-3 minutes until the potatoes start to absorb the flavors.
Pour in the vegetable broth, then bring the mixture to a gentle simmer. Cover the pot partially and cook for about 20-25 minutes, or until the potatoes are tender when pierced with a fork and the soup looks invitingly warm and slightly thickened.
Use an immersion blender directly in the pot to blend the soup until completely smooth and velvety. Alternatively, carefully transfer the hot mixture in batches to a blender, then return it to the pot.
Stir in the heavy cream and warm the soup gently for 2 minutes. Taste and season with salt and freshly ground black pepper as needed. For added richness, stir in a tablespoon of butter at this stage.
Once everything is well combined and heated through, turn off the heat. Let the soup rest for 5 minutes to allow flavors to meld and the texture to thicken slightly.
Spoon the velvety soup into bowls, garnish with chopped herbs if desired, and enjoy its warm, soothing aroma and silky texture.
