Heat the butter in a soup pot over medium heat. Add the garlic, leek, celery, and half of the ginger. Cook, stirring occasionally, until the leek and celery are softened, 8 to 10 minutes.
Add the broth, pumpkin, sweet potato, cinnamon stick, and nutmeg. Bring to a simmer and cook until the pumpkin is very tender, about 30 minutes.
Remove and discard the cinnamon stick. Purée the soup until quite smooth. Strain through a fine sieve for an exceptionally smooth texture, if you wish.
Return the soup to medium heat. Add the wine, milk, lime juice, and the salt to taste. Stir to combine well and reheat the soup to just below a simmer.
Whip the chilled heavy cream to medium peaks and fold in the remaining ginger. Serve the soup in heated bowls, garnished with a dollop of ginger-flavored cream.