In a food processor, pulse chestnuts until nearly smooth. In a small saucepan, combine sugar, water, corn syrup and salt. Brush down the sides of the pan with a wet pastry brush to dissolve any sugar granules that may be stuck. Cook sugar over medium heat until it reaches 240°F; about 6-8 minutes. Remove from heat. Immediately pour the hot sugar through the food processor feed tube while processing the chestnuts. Continue processing until completely smooth. Transfer to small bowl; cover with plastic wrap. The chestnut cream will thicken as it cools.
For cookies, combine flour, cocoa powder, baking powder and salt in small bowl; set aside.
In a large bowl, beat butter and sugar with electric mixer until light and fluffy. Add egg; mix well. Add cooled chocolate and vanilla and beat until well combined. Add flour mixture, 1 cup at a time, scraping bottom and sides of bowl and mixing well after each addition. Divide dough into 2 disks. Wrap tightly in plastic wrap and refrigerate at least 1 hour, or until firm enough to roll.
Preheat oven to 350°F.
On a floured surface, roll each dough disk about 1/8 in. thick. Cut into 2 in. rounds. Cut a 1 in. round in the center of half of the cookies; these will be the tops. Gather scraps, reroll, and cut additional cookies. Bake cookies on an ungreased cookie sheet 6-8 minutes or until cookies are fragrant. Cool 3-4 minutes; remove to cooling grid and cool completely.
To assemble, spread a thin layer of chestnut cream on whole cookies. Arrange cut out cookies on cookie sheet; dust with confectioners' sugar. Sandwich cookies together. Cookies soften quickly after they are filled.