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Melt butter over medium heat in large soup pot. Add leeks and garlic and cook 10 minutes, stirring regularly, until soft and wilted. Adjust heat as necessary so as not to brown.
Add potatoes, broth, sea salt, pepper, and thyme, if using, to pot and bring to boil. Cover and turn heat low. Simmer 15 minutes, or until potatoes are very soft.
Purée soup with hand-held immersion blender until smooth. (Alternatively, use a standard blender to purée soup in batches.) Add heavy cream and bring to simmer. Taste and adjust seasoning with sea salt and pepper. If soup is too thin, simmer until thickened. If too thick, add water or stock to thin it out. Garnish with chopped chives, if desired.