Preheat your oven to 325 degrees F. Place the beets in a bowl and sprinkle with 1T oil and season with salt. Toss to coat. Place the beets on half of a large sheet of aluminum foil, and fold over to make a pouch. Seal tightly. Place in the oven and roast until a fork easily pierces the beets, about 1 hour and 15 minutes. Remove from the oven and let cool. While the beets are still slightly warm, remove the outer skin by rubbing them gently with a towel. Cut the beets into quarters.
In a large soup pot, warm the remaining olive oil over medium heat. Add the leek and cook until it begins to soften, stirring occasionally, about 5 minutes. Flick in the garlic and ginger; cook for 2 minutes more. Season lightly with salt. Add the beets, chicken broth, and 3 cups of water. Bring to a simmer and cook for 10 minutes.
Transfer the soup in batches to a food processor and puree until smooth. Then return the soup to the pot and place over medium heat. Taste and adjust seasoning, adding more salt and lemon juice to taste and more ginger if you wish. Just before serving (the soup should not be served too hot; it could even be served chilled or at room temperature), add the cucumber to the soup and stir. Ladle into bowls. You may serve this with a dollop of creme fraiche.