Trim the dark green tops from the leeks, then slice the white and light green parts into thin half-moons. Rinse thoroughly in cold water to remove any grit, then drain well.
Peel the potatoes with a sharp peeler, then cut them into roughly 1-inch chunks for even cooking.
Place a large heavy-bottomed pot over medium heat and add the butter. Once melted and fragrant, add the sliced leeks. Sauté for about 5-7 minutes, stirring occasionally, until they soften and release a sweet aroma, turning a light golden color.
Add the chopped potatoes to the pot and stir to coat them with the butter and leek mixture. Cook for another 2 minutes, allowing the flavors to meld.
Pour in the warmed broth, then bring the mixture to a gentle simmer. Reduce the heat to low and cover loosely with a lid. Let it simmer for about 20-25 minutes, until the potatoes are tender and can be mashed easily with a fork.
Turn off the heat, then use an immersion blender directly in the pot to blend the soup until smooth and velvety. Alternatively, transfer the soup in batches to a blender, then return it to the pot.
Stir in the heavy cream, then warm the soup gently over low heat, stirring constantly to prevent curdling. Taste and season with salt and freshly cracked pepper as needed.
Let the soup rest for a few minutes to allow flavors to meld, then ladle into bowls. Garnish with chopped herbs if desired and serve hot, enjoying the silky texture and comforting aroma.
