One day in 1832 when Austrian diplomat Klemens von Metternich demanded that his kitchen create a dessert to impress visiting dignitaries, his chef was out sick. Franz Sacher, a 16-year-old kitchen apprentice, used what he had on hand to create what would become the most well-known Viennese dessert: a dark chocolate sponge cake with apricot jam.
Make the cake: Preheat the oven to 325 degrees F. Butter the bottom and sides of an 8-inch springform pan; line the bottom with parchment paper. Butter the parchment and dust the bottom and sides with cake flour, tapping out the excess. Microwave the semisweet chocolate in a microwave-safe bowl, stirring every 30 seconds, until smooth; let cool.
Meanwhile, sift the cake flour and salt into a medium bowl. Beat the butter and 3/4 cup confectioners' sugar in a large bowl with a mixer on low speed until combined. Increase the speed to medium high and beat until light, 1 to 2 minutes. Add the 6 egg yolks, one at a time, then beat in the vanilla. Reduce the mixer speed to low and beat in the melted chocolate. Add the flour mixture and beat until just combined.
Beat the 4 egg whites in a separate large bowl with a mixer on medium speed until frothy, about 30 seconds. Add the remaining 2 tablespoons confectioners' sugar. Increase the speed to high and beat until the egg whites are stiff and glossy, about 1 more minute. Working in two batches, fold the egg whites into the batter with a rubber spatula until just combined. Spread the batter in the prepared pan and bake until a toothpick inserted into the center of the cake comes out clean, 40 to 45 minutes. Transfer to a rack and let cool 10 minutes, then remove the springform ring and let the cake cool completely. Remove the cake from the bottom of the pan and cut in half horizontally using a long serrated knife.
Make the filling: Combine 1/2 cup apricot jam, the granulated sugar and water in a small saucepan. Bring to a simmer, stirring, until the jam melts; strain through a fine-mesh sieve into a medium bowl and stir in the rum. Gradually brush about two-thirds of the syrup on the cut sides of the cake, allowing it to soak in. Remove any large pieces of fruit from the remaining 1/2 cup apricot jam; spread over the cut side of the bottom cake half and top with the remaining cake half, cut-side down. Brush the top and sides of the cake with the remaining syrup, allowing it to soak in.
Make the glaze: Combine the corn syrup and water in a small saucepan and bring to a simmer over low heat, stirring to combine. Remove from the heat and add the bittersweet chocolate; stir until melted and smooth.
Transfer the cake to a rack set over a baking sheet. Working in two or three batches, pour the glaze over the cake, allowing the excess to drip down the sides. Smooth the glaze with an offset spatula if needed. Refrigerate until the glaze is set, about 30 minutes. Serve with whipped cream.