1. Preheat oven to 350 degrees. Coat a 9-inch round cake pan with cooking spray. Line bottom with parchment cut to fit, and coat parchment with spray. Dust with flour, and tap out excess.

2. Whisk together flour, baking powder, and salt in a medium bowl.

3. Using the tip of a paring knife, scrape seeds from vanilla bean into a large bowl, and discard pod. Add 1 cup sugar, and beat with a mixer on medium-high speed for 30 seconds. Add butter, and beat until pale and fluffy. Beat in egg yolks. Reduce speed to low. Add flour mixture in 3 additions, alternating with chestnut puree after first addition and milk after second, scraping down sides of bowl as needed. Transfer batter to a large bowl.

4. Beat egg whites in a clean bowl with a mixer on medium-high speed until soft peaks form. Gradually add remaining 1/2 cup sugar, beating until stiff, glossy peaks form. Whisk one-third of the egg whites into batter just until combined. Gently but thoroughly fold in remaining whites. Scrape batter into prepared pan, and smooth top.

5. Bake until a toothpick inserted into center of cake comes out clean, about 1 hour. Let cool completely in pan on a wire rack. (Cake can be wrapped in plastic and stored at room temperature for up to 2 days.)

6. Using a serrated knife, trim rounded top of cake to create a flat surface. Slice cake horizontally into 3 layers. Place bottom layer on a cake stand, and brush generously with Armagnac syrup. Spread 1 1/4 cups pastry cream over syrup, leaving a 1/4-inch border around edge. Top with second cake layer, syrup, and pastry cream. Top with remaining cake layer, bottom up. Refrigerate for 30 minutes.

7. Spread frosting over top and sides of cake. Refrigerate for 10 minutes. Pour glaze over top, letting some drip down sides. Garnish with roasted chestnuts if desired.