Optional: Make the leek top broth. Cut the dark green tops off the leeks. Wash well and add to a large pot with the water, thyme, garlic, salt, and peppercorns. Bring to a boil over high heat, then reduce the heat and simmer for 30 minutes. Strain and measure 6 cups broth for the soup. Freeze the remaining broth or store it in the refrigerator for another use.

Make the soup: Chop the white and light green parts of the leeks. Wash well.

Heat the butter in a large pot or Dutch oven over medium heat. Add the leeks, salt, and several grinds of pepper and cook for 8 to 10 minutes, stirring occasionally, or until the leeks are soft. If the leeks start browning before they soften, reduce the heat.

Add the potatoes, garlic, broth, and thyme and simmer for 20 minutes, or until the potatoes are very soft. Remove the thyme, then use an immersion blender or transfer the soup to an upright blender to blend until smooth. Return the soup to the pot and stir in the milk. Season to taste, adding the lemon juice, if desired.

Garnish with chives and serve.