Preheat the oven to 375 degrees F (190 degrees C) with a rack in the center of the oven. Butter and flower the sides of 9-inch springform pan, then line the bottom with a circle of parchment paper.
Place the Bittersweet Chocolate (3/4 cup) in a heat-proof bowl and heat over a small saucepan of barely simmering water or in the microwave until just melted. Set aside and cool completely while stirring often.
Place Unsalted Butter (9 tablespoon) in a large mixing bowl and beat on medium speed until very light and creamy. Add the Powdered Confectioners Sugar (1 cup) on low speed then increase to medium speed and beat again until light and creamy.
Separate the yolk from white of the Egg (6) Beat the egg yolks in, one at a time, into the butter mixture.
Add the cooled, melted chocolate and Vanilla Extract (1 teaspoon) into the egg butter mixture. Scrap down the sides of the bowl and mix again.
In a clean bowl, beat the egg whites with about one tablespoon of the Granulated Sugar (1 1/2 cup) on high speed until foamy. Gradually add in the rest of the sugar and continue beating the whites until they form soft peaks.
Stir about 1/3 of the whipped egg whites into the chocolate mixture, then gently fold the remaining egg whites into the chocolate mixture with a spatula until just a few wisps of egg white remain.
Stir together the All-Purpose Flour (1 cup) and Fine Salt (1 pinch) and sift half of it over the chocolate mixture. Fold in with a spatula until almost incorporated. Sift the remaining flour and fold to combine completely.
In a prepared springform pan, spread the batter evenly and bake for 35-45 minutes or until a toothpick comes out clean. Meanwhile, make the apricot glazing.
To make the apricot glaze, boil the Apricot Jam (1 1/4 cup) and Water (2 tablespoon) in a small saucepan over medium heat. Cook and stir until the mixture thickens, about 2-3 minutes. Strain through a wire mesh sieve and keep warm until use.
Cool the cake in the pan for 10 minutes, then run a knife around the edge to loosen it from the pan and remove the sides. Then carefully remove the pan and allow cake to cool right-side-up on a rack.
Turn the cake upside-down so that the flat side is on top. Cut the cake horizontally into two even layers.
Place one cake layer on the 8 1/2-inch cardboard cake round and spread it generously with about half of the apricot glaze. Allow it to soak in.
Place the second cake layer on top and spread the top and sides with remaining apricot glaze. Work quickly before the glaze has a chance to set and use a metal offset spatula to smooth the top. Allow the apricot glaze to set once it's smooth.
To make the chocolate glaze, place the Granulated Sugar (1 cup) and Water (1/2 cup) in a heavy-bottomed saucepan over medium-high heat and bring to a boil, stirring occasionally.
Attach a candy thermometer and cook, stirring, until the mixture reaches 234 degrees F or 112 degrees C, about 5 minutes. Remove the saucepan from the heat and whisk in the Bittersweet Chocolate (4 ounce). If it thickens, return it to low heat and add few drops of water if necessary.
This glaze should be smooth and shiny. Turn off heat and stir the glaze for 30-60 seconds to cool it slightly, then immediately use it to glaze the cake.
Pour the chocolate glaze over the top of the cake, first around the edges then in the middle. Spread the excess glaze over any bare spots using a metal offset spatula.
If you want extra chocolate on your cake, you can whip up the ganache by pouring warm Cream (200 milliliter) over Bittersweet Chocolate (200 gram). Leave it for 3-5 minutes and stir the chocolate and cream until well mixed. Cool for 10 minutes and pour over glazed cake.
To make the writing chocolate, heat Bittersweet Chocolate (1/4 cup) until just melted and stir in Vegetable Oil (1/2 teaspoon) to get a pipeable consistency. Let the mixture cool down a little if it's too runny. Place the mixture into a disposable piping bag.
With the writing chocolate, pipe the word "Sacher" in the middle of the cake and add any decorative flourishes you wish. Chill the cake until the glaze is completely set, about an hour.
To serve, let the cake come to room temperature for about one hour before serving. Whip the cream to soft peaks. Cut the Sachertorte into wedges with a large sharp knife dipped in hot water and wipe off the blade between cuts. Serve each slice with a dollop of whipped cream.