Melt chocolate in top of a double boiler over simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching. Let cool slightly, about 10 minutes.
Pour aquafaba into the bowl of a stand mixer fitted with a whisk attachment. Beat on high speed until fluffy and quadrupled in volume, about 1 minute. Add xanthan gum and beat for 30 seconds. Add confectioners’ sugar and vanilla sugar; continue beating until foam is firm and glossy, about 2 minutes more.
Fold melted chocolate gently into whipped foam until thoroughly incorporated. Transfer to a lidded container.
To make the cake: Heat oven to 200 degrees F. Line two baking pans with parchment paper and set aside. Draw two 9-inch circles in the center of the paper and set aside.
Beat egg whites and cream of tartar until foamy. Stream in 1 1/4 cups sugar and continue to beat on high speed. Just as the whites reach their peaks, sprinkle in sifted cocoa and cayenne pepper and continue beating to stiff peaks.
Spread the meringue within the circles. Bake for 3 hours, turn off oven, and let sit for 1 hour.
Cool on the pan on a wire rack. Store in a very dry, cool place.
Fill the layers with whipped cream and serve within 2 hours.