Preheat oven to 350. Place chestnuts flat side down on a cutting board; slice an X through the outer layer of each chestnut shell. Place these in a single layer on a baking sheet and put them into the oven.
After about 5–10 minutes, the shell should start to peel back from the nut. Remove the nuts from the oven and peel them. (They will be very hot, so be careful not to burn your fingers.) Note: If you are using frozen chestnuts, you can skip this step and simply cut the defrosted chestnuts in half and remove the meat. Freezing the chestnuts first causes the meat to pull away from the center of the shell.
In a heavy-bottomed pan, place the chestnuts and enough water to cover them. Over medium heat, cook the chestnuts for 35–40 minutes, until they are fork tender. Drain them and then put them through a food processor until they form a paste.
Return the chestnuts to the pan, add the sugar, vanilla bean, water and salt. Bring the mixture to a simmer and cook over low heat. Stir every few minutes to prevent the mixture from burning on the bottom. Cook for 30–40 minutes, until the mixture has reduced and formed thicker syrup. When you run a spoon around the edge, you should be able to briefly see the bottom of the pan.
Cool. Remove the vanilla bean and store the puree in clean jars in the refrigerator.