In a medium saucepan, melt the butter. Add the onion and cook over moderate heat until softened, about 7 minutes. Add 3 cups of the stock and simmer over low heat for 20 minutes. Add the edamame and cook until tender, about 10 minutes. Let cool slightly, then puree the soup in a blender until very smooth. Transfer the soup to a bowl and refrigerate until chilled, at least 3 hours.

Whisk the remaining 1 cup of chicken stock into the soup and season with salt. In a small bowl, combine the creme fraiche with the grated ginger. Ladle the soup into bowls, drizzle with the ginger creme fraiche and serve.