Step 1 Put the milk, 4 cups of water, the coarse salt, celery root and potatoes in a large pan and bring to a boil over medium heat. Lower the heat and cook at a simmer until the vegetables can be easily pierced with the point of a knife, 20 to 25 minutes. Skim off foam as necessary. Drain the vegetables and return them to the pan.
Step 2 Put the pan back over low heat and toss the vegetables just enough to cook off their excess moisture, 1 to 2 minutes. Transfer the vegetables to the work bowl of a food processor. Add the butter and pulse, taking care not to overwork the mixture, just until the puree is smooth and creamy. Season with salt and pepper. Keep the puree warm in the top of a double boiler over simmering water.