Clean leeks, trim and discard green tops and white roots. Split leeks lengthwise and place in plenty of water. Swish the leeks in the water to release any sand or soil. Drain. Repeat until no grit remains.

Slice leeks, white and light green parts only

Heat oil in a Dutch oven over medium heat. Add leeks and 1/4 teaspoon salt. Cook, stirring occasionally, until very tender but not brown, 20 to 30 minutes. Adjust heat as necessary and add a bit of water if needed to prevent sticking. Set aside 1/3 cup of the leeks in a small bowl.

Add potatoes, milk and broth to the pot. Bring to a simmer (do not boil) and cook, stirring occasionally, until the potatoes are very tender, 10 to 15 minutes.

When the potatoes are tender, remove from the heat. Puree the soup with an immersion blender or in batches in a regular blender.

Season with the remaining 1/2 teaspoon salt, lemon juice and white pepper. Divide among 8 soup bowls and float a crouton on top of each. Garnish with chives, if desired.