Begin by trimming the roots and dark green tops of the leeks, leaving the white and light green parts. Slice them lengthwise, then chop into thin slices. Rinse thoroughly under cold water, separating the layers to remove any dirt or grit. Drain well.
Heat the olive oil in a large pot over medium heat. Add the cleaned leeks and cook gently for about 10 minutes, stirring occasionally. The leeks should soften and become fragrant without browning.
Add the diced potatoes to the pot, stirring to combine. Pour in the chicken broth and bring the mixture to a boil. Reduce heat and let it simmer for 20 minutes or until the potatoes are fork-tender.
Once the potatoes are soft, use an immersion blender to puree the soup directly in the pot until smooth and creamy. If you don’t have an immersion blender, carefully transfer the soup in batches to a regular blender and blend until smooth.
Lower the heat and stir in the heavy cream. Season with salt and pepper to taste. Warm the soup gently for another 5 minutes, but don’t let it boil.
Ladle the soup into bowls and sprinkle with fresh chopped chives. Serve immediately with crusty bread or your favorite side.