For candied chestnuts, preheat oven to 400 °F. Score a line through the shell of each chestnut and place on a baking tray. Bake for 15 to 20 minutes, until chestnut skins split open. Cool until no longer warm, but not completely cooled. Peel chestnuts.
Bring sugar and water up to a simmer and add chestnuts. Simmer for 15 minutes, then drain. Let chestnuts dry on a cooling rack.
For sponge, reduce oven temperature to 375°F and grease and flour a 10-inch spring form pan, tapping out excess flour. Warm eggs in their shells in hot tap water for 5 minutes. Whip warmed eggs and ½ cup sugar on high speed for 15 minutes. Sift flour and while mixer is on low speed, add in over 5 seconds. Increase speed to high once, then stop. Scrape batter into prepared pan and bake until top of cake springs back when touched, about 20 minutes. Cool completely.
Wash spring form pan and re-line with parchment paper. Place cooled sponge cake in pan. Heat remaining 2 Tbsp sugar and rum until sugar is dissolved. Brush over sponge cake and set aside while preparing chestnut cream.
For chestnut cream, heat milk and scraped seeds from vanilla bean until just below a simmer. Whisk egg yolks and sugar and gradually pour in hot milk, whisking constantly, until all liquid has been added. Return to stove and stir with a wooden spoon over medium heat until it coats the back of a spoon, about 4 minutes. Strain, cover surface with plastic wrap and cool.
Sprinkle gelatin over rum and stir. Set aside.
In a food processor, pulse chestnut purée and cooled custard until smooth. Measure 1 cup of purée and heat over low heat until warm. Stir in gelatin until melted and add back chestnut mixture to base. Whip cream to medium peaks and fold into chestnut base. Pour and spread over sponge cake and chill for at least 4 hours.
To garnish, remove ring from spring form pan, peel away parchment and slide cake onto a serving plate. Arrange candied chestnuts around outer edge on top of cake. Dust center of cake with cocoa powder.