Set aside 2 tablespoons of the butter in the refrigerator. Heat the remaining butter in a large soup pot over medium heat. Add the onion, leeks, and garlic and cook, covered, stirring occasionally, until soft, about 12 minutes. Raise the heat to high, pour in the vermouth, and cook, uncovered, until almost dry, about 4 minutes.
Add the potatoes, water, and salt. Tie the parsley, thyme, and bay leaf together with a piece of kitchen twine and add to the soup. Bring to a boil, lower the heat, and simmer, with the cover slightly ajar, until the potatoes are very tender, about 20 minutes. Remove from the heat and allow to cool.
When the potato mixture has cooled, remove and discard the herb bundle. Working in batches, transfer the mixture to a blender and puree until smooth, or puree with an immersion blender. Return the puree to the pot and re-heat over medium heat.
Whisk in the reserved 2 tablespoons butter, heavy cream, and lemon juice into the soup and season with pepper, to taste. Divide among warm soup bowls, garnish with the chives, and serve immediately.
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