Melt butter over medium heat in a large soup pot. Add leeks and garlic and cook, stirring regularly, until soft and wilted, about 10 minutes. Adjust heat as necessary so as not to brown.
Add potatoes, broth, sea salt, pepper, and thyme, if using, to pot and bring to a boil. Cover and turn heat low. Simmer for 15 minutes, or until potatoes are very soft.
Remove thyme sprig, then purée soup with a hand-held immersion blender until smooth. (Alternatively, use a standard blender to purée the soup in batches.) Add heavy cream and bring to a 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 fresh herbs, if desired.