Peel the potatoes and cut them into 1 1/2-inch chunks. Cover with water in a large pot, add some salt, bring to a boil, and cook until very tender, about 20 minutes. Drain.

Peel the celery root and cut it in 1-inch dice. Put it into a saucepan with the butter and enough water to barely cover. Cook, covered, for 20 to 25 minutes, until the celery root is tender. If necessary, add additional water to keep the celery root from sticking to the pan. The celery root will be nearly pureed.

Warm the milk or cream in another saucepan. Pass the celery root and the potatoes through a food mill, ricer, or food processor. Add enough warmed milk or cream for the puree to reach the consistency you want. Season with salt and pepper and a splash of white wine vinegar to heighten the flavor of the celery root.