Combine chestnuts, milk, and salt in a saucepan. Using the tip of a paring knife, scrape seeds from vanilla bean into pan, and toss in pod. Bring to a simmer, and cook, partially covered, until chestnuts are very soft and all but about 1/2 cup of the liquid has been absorbed, about 20 minutes.
Pour mixture through a fine-mesh sieve into a bowl, reserving chestnuts and cooking liquid separately and discarding vanilla pod. Process chestnuts in a food processor until very smooth, adding reserved cooking liquid (and additional milk if needed), 1 tablespoon at a time, until mixture is the consistency of a thick spread. Pour through a large-mesh sieve. Cover with plastic wrap, and refrigerate until ready to use.