In a large saucepan, bring vegetable broth to a simmer. Add beets and garlic. Cover and simmer for 45 minutes or until beets are tender. Season with salt. Transfer beets and liquid to a blender and blend until smooth. Add butter and blend until butter is melted and blended into soup. Keep mixture warm until ready to serve.
To serve, ladle beet soup into shallow bowls. Spoon pasta evenly into each dish, then top with beet greens. Sprinkle goat cheese and pecans on top. Finish with fresh ground pepper and serve.