Line a baking sheet with aluminium foil. Place beet on top and sprinkle with water. Cover with aluminium foil loosely and bake in preheated moderate oven to 350ºF (180ºC) for approximately 40 minutes or until beets are tender when pierced with a fork. Remove from the oven, cool on the sheet and peel.
Slice beets in very thin slices (with a knife or a slicer). Place slices on a serving platter or plates, slightly overlapping. Sprinkle over with goat cheese and chopped onion. Season with salt and pepper.
Whisk vinegar, mint leaves, oil and sugar in a small bowl. Season with salt and pepper and pour over carpaccio. Garnish with chives.