Prepare the leeks: Trim off the root end and dark green tops of the leeks, then slice them in half lengthwise. Rinse the leeks under cold water to remove any dirt or grit trapped between the layers. Pat them dry with paper towels and thinly slice.

Cook the leeks: In a large pot or Dutch oven, melt the butter or heat the olive oil over medium heat. Add the sliced leeks to the pot and cook, stirring occasionally, until they are softened but not browned, about 5-7 minutes.

Add the potatoes: Add the diced potatoes to the pot with the leeks and stir to combine.

Simmer the soup: Pour in the vegetable or chicken broth and water, making sure the potatoes are submerged. Season with salt and pepper to taste. Bring the soup to a simmer, then reduce the heat to low. Cover the pot and let the soup simmer gently for about 15-20 minutes, or until the potatoes are tender when pierced with a fork.

Blend the soup: Once the potatoes are cooked through, use an immersion blender to blend the soup until smooth and creamy. Alternatively, carefully transfer the soup in batches to a blender and blend until smooth, then return it to the pot.

Finish the soup: If desired, stir in the heavy cream or milk to add creaminess to the soup. Adjust the seasoning with salt and pepper to taste.

Serve: Ladle the Leek and Potato Soup into bowls and garnish with chopped fresh chives or parsley, if desired. Serve hot and enjoy!

This Leek and Potato Soup is simple to make and packed with comforting flavors. It's perfect for a cozy meal at home or for sharing with loved ones on a chilly day.