Trim leafy tops off beets, leaving 1 in. of stems attached. Wash beets and leaves, discarding tough stems. Put beets in a medium saucepan and cover with water. Bring to a boil, covered; reduce heat and simmer until beets are tender when pierced, 35 to 45 minutes. Drain and let cool. Remove skin and stems, then cut beets thinly crosswise. Cut beet leaves crosswise into thin shreds. Set aside.
Heat oil in a large pot over medium-high heat. Add onion and garlic and cook until lightly browned, about 4 minutes. Add broth and sliced beets. Bring to a boil. Stir in reserved beet greens and cook until wilted, about 1 minute. Add more broth if you like. Season with salt and freshly ground black pepper.
Note: Nutritional analysis is per serving.