Trim off the dark green tops of the leeks, then slice the white and pale green parts thinly. Rinse thoroughly in cold water to remove any grit, then set aside.
Peel the potatoes and chop them into roughly 2 cm cubes to ensure even cooking.
Heat the butter in a large heavy-bottomed pot over medium heat until it shimmers and begins to melt softly.
Add the sliced leeks to the pot and cook gently, stirring occasionally, until they turn translucent and fragrant—about 8-10 minutes. The aroma should become sweet and earthy, and the leeks will soften beautifully.
Stir in the chopped potatoes, coating them in the buttery leek mixture, and cook for another 2 minutes to meld the flavors.
Pour in the vegetable broth, bringing the mixture to a gentle simmer. Partially cover the pot and cook until the potatoes are fork-tender—about 15-20 minutes. The broth should bubble softly, and the potatoes will become very soft and easily mashable.
Turn off the heat, then use an immersion blender to blend the soup directly in the pot until smooth and velvety. Alternatively, carefully transfer the soup in batches to a blender and puree until silky.
Stir in the heavy cream, then taste and season with salt and pepper as needed. Let the soup rest for a few minutes to allow flavors to meld and develop a rich, smooth texture.
Spoon the warm, silky soup into bowls, garnishing with a drizzle of oil, chopped chives, or crispy bacon if desired. Serve immediately for the best flavor and texture.
