Saute garlic in butter in a 5 to 6 quart pan. Add broth, potatoes, onion, celery, parsley, green onion, and thyme. Cover and bring to a boil over high heat, stirring occasionally. Reduce heat and simmer, stirring occasionally, until potatoes mash easily with a fork, 20 to 25 minutes. Remove from heat and let cool to room temperature, about 1 hour.
Meanwhile, thinly slice a half of one cucumber; wrap airtight and chill. Coarsely chop remaining.
Working in batches, whirl potato mixture and chopped cucumber in a blender until smooth; pour into a large bowl. Stir in creme fraiche, salt, and hot sauce. Cover and chill until cold, at least 1 hour.
Thin soup with more broth if necessary. Test for salt and hot sauce and ladle into bowls and garnish with sliced cucumber. Serves 8.