Place the beets in a pot and cover with cold water. Set on the stove and gently boil until tender, about 30 minutes. Cool the beets in cold water. Peel the beets, cube them, and set in a bowl.
Heat the oil in a large to medium pot set over medium heat. Add the onion and sliced fennel and sauté until tender, about 5 minutes. Add the stock and cubed beets and bring to a simmer. Simmer the soup 10 minutes.
Purée the soup in the pot with an immersion blender; or purée in a food processor or blender. Return the soup to a simmer. Season the soup with salt and pepper.
Ladle soup into bowls. Drizzle each soup with some balsamic vinegar. Top with feta cheese and chopped fennel fronds and serve.
Note: One quite small fennel bulb should yield the 1 cup needed for this recipe. To prepare the fennel, trim off the stem-like top and save some of the leafy fronds for the soup’s garnish. Peel the outside of the fennel bulb if it appears to be tough or is discoloured. Remove the centre core from the bulb. Now cut the fennel bulb in half and it’s ready to thinly slice. If you have any leftover fennel, keep it and add it to a salad or other dish the next day.