Lightly coat an 8-inch round cake pan with oil, and line with plastic, leaving a 4-inch overhang.
Finely grind chestnuts in a food processor. Heat ground chestnuts, milk, 3 tablespoons sugar, and the salt in a medium saucepan over medium-high heat, stirring occasionally, until bubbles form at edge.
Whisk together yolks, cornstarch, and remaining 2 tablespoons sugar in a medium heatproof bowl.
Slowly add hot chestnut mixture to yolk mixture, whisking constantly. Return to pan; cook over medium-low heat, whisking constantly, until thick, about 2 minutes. Transfer to the bowl of a mixer. Add vanilla and butter. Beat on medium speed until cool, about 5 minutes. Stir half the chestnut mixture into melted chocolate.
Beat 1 1/4 cups cream until soft peaks form. Using a rubber spatula, gently fold half the whipped cream into plain chestnut mixture and the other half into chocolate-chestnut mixture.
Pour chestnut mousse into prepared pan. Cover with 12 chocolate wafers, working in a circular pattern. Top with chocolate-chestnut mousse and remaining 12 chocolate wafers. Fold plastic overhang over top to cover cake. Refrigerate overnight.
Uncover cake; place a platter on top, flip, and unmold. Remove plastic. Beat remaining 1/4 cup cream until soft peaks form. Spread on top of cake. Garnish with chocolate shavings.