Slice the cleaned leeks thinly, exposing the light green and white parts. Rinse thoroughly under cold water to remove any dirt or grit, then drain and set aside.
Heat the saucepan over medium heat and melt the butter until it begins to foam and turn lightly golden. Add the sliced leeks and cook, stirring occasionally, until they soften and turn translucent, about 5-7 minutes. The leeks should develop a slight caramelization around the edges.
Add the diced potatoes to the pot with the leeks, stirring to coat for 1-2 minutes. Pour in the broth, bringing the mixture to a gentle simmer. Cover the pot partially and cook until the potatoes are tender, about 15 minutes. You should see the broth bubbling gently, and the potatoes becoming easily pierced with a fork.
Once the potatoes are tender, use an immersion blender to puree the soup directly in the pot until smooth and velvety. Alternatively, transfer the mixture in batches to a blender, then return to the pot. The soup should have a uniform, creamy texture with no visible chunks.
Taste the soup and season with salt and freshly ground pepper as desired. If the soup appears too thick, add a little more broth or water to achieve your preferred consistency. Warm the soup through on low heat for a few minutes, allowing flavors to meld.
Serve hot in bowls, garnished with a sprinkle of freshly chopped herbs or a drizzle of olive oil if desired. The finished dish is smooth, pale greenish-white, and has a silky texture with a delicate leek onion flavor.
