Combine chestnuts, broth, water, thyme and salt in a medium saucepan. Bring to a simmer over medium heat and cook, partially covered, until the chestnuts easily break apart, 10 to 20 minutes. Remove the thyme.
Transfer the chestnut mixture to a food processor; puree until the mixture is very smooth and has the consistency of a thick spread. Add a little more broth if a thinner consistency is desired. Return the puree to the pan; cover to keep warm.
Reheat over low heat just before serving; thin with a little water, if desired.
Make Ahead Tip: Prepare through Step 2 up to 1 hour in advance.
Tip: Cooked peeled chestnuts are available in jars in the baking aisle or with seasonal foods at many supermarkets.
3 1/2 starch