Shredding the beets and apples for this deep fuchsia soup cuts down on the cooking time and keeps the sweet-tart flavors fresh. Slideshow: More Healthy Soup Recipes
In a food processor fitted with the shredding disk or using a box grater, shred the beets and apples.
In a large saucepan, heat the olive oil. Add the shallots and garlic and cook over moderately high heat, stirring occasionally, until softened, about 3 minutes. Add the shredded beets and apples and cook, stirring occasionally, until just starting to soften, about 5 minutes. Add the stock and 3 1/2 cups of water and bring to a boil. Simmer over moderate heat, stirring occasionally, until the beets are tender, 25 to 30 minutes.
In a blender, puree the soup with the apple juice until very smooth. Transfer to a bowl; refrig-erate until chilled, 45 minutes.
Stir tablespoons of water into the soup if it gets too thick, then season with salt. Ladle the soup into bowls and swirl in the sour cream. Garnish with shaved or julienned beets and beet greens and serve.
The soup can be refrigerated for up to 3 days.