Step 1Combine the chestnuts, celery root and shallot in a large saucepan. Add the chicken broth and 1 1/4 cups water. There should be enough liquid to just cover the chestnuts and celery root, but if necessary add a little more water. Stir in the salt.
Step 2Bring to a simmer, cover and cook until the chestnuts and celery root are tender enough to crush against the side of the pan, about 20 minutes.
Step 3Use a slotted spoon to transfer the cooked chestnuts and celery root to a food processor and puree. Scrape down the sides of the work bowl, add the butter and puree until smooth. Serve immediately. This makes about 3 cups puree.