Warm the olive oil in a heavy-bottomed pan over medium low heat. Cook the onion in the olive oil over low heat without letting it brown for 15 minutes. Add the cauliflower, salt to taste, and the water. Raise the heat, cover the pot tightly, and simmer the cauliflower for 20 minutes, or until tender.
Working in batches, purée the soup in a blender to a very smooth, creamy consistency. Let the soup stand for 20 minutes. In this time it will thicken slightly.
Thin the soup with 1/2 cup hot water. Reheat the soup. Serve hot, drizzled with a thin stream of extra-virgin olive oil and freshly ground black pepper. Sprinkle with fresh thyme leaves, crushing between fingers to release the smell and flavor.