Bring milk, water, and salt just to boil in heavy large saucepan over high heat. Add celery root cubes, potato cubes, and onion quarters; bring to boil. Reduce heat to medium and simmer until vegetables are tender, about 30 minutes. Drain, discarding cooking liquid.
Combine vegetables and butter in processor and puree until smooth. Season to taste with salt and white pepper. Do ahead Celery root puree can be made 1 day ahead. Cover and chill. Rewarm in microwave before serving.
Transfer celery root puree to bowl. Sprinkle with chopped fresh chives and serve.