Preheat the oven to 175 C. Grease an 8-inch springform tin and set aside.
Make the sponge cake. Melt the chocolate in the microwave in 10-second increments or in a double boiler until smooth. Don't overheat the chocolate, stir to melt down any lumps. Set aside to cool.
In a mixing bowl, beat the butter and half of the sugar (1/4 cup) with a hand mixer on medium speed until pale and creamy. Add the vanilla and beat to combine.
Now separate the eggs, adding the yolks to a small bowl, and the whites to another larger mixing bowl or stand mixer bowl for later. Take care that no yolk gets in with the whites.
To the butter and sugar, add the yolks one at a time, beating briefly after each to combine. Add the melted chocolate and beat again to combine until creamy.
Now beat the whites (either with the same rinsed and dried hand mixer or use a stand mixer with the whisk attachment) for 1 minute on medium-high speed until foamy. Add the remaining 1/4 cup sugar and mix on high for 3 to 4 minutes until stiff peaks form when you lift out the beater. See photos below.
Fold the whipped egg whites into the chocolate mixture in three additions. Be gentle so the whites don't deflate entirely, fold in one direction only and use a silicon spatula for ease.
Finally, add the sifted flour in two additions and fold again to combine gently until no lumps remain. In the absence of baking powder, a sponge relies on air beaten into the batter to help the cake rise properly. So don't beat it too vigorously.
Transfer the batter to the prepared pan, smoothen the top and bake for 30 to 35 minutes until a toothpick poked in the center comes out clean. Set aside to cool for half an hour, then remove the sides of the springform pan and let the cake cool completely at room temperature. It will be about an inch tall, it's not a very large cake. The top may look slightly 'crumpled' because the cake falls a little as it cools.
Make the jam layer. Add 1/3 cup of the apricot jam (remove any seeds) into a small saucepan along with the rum. Heat on low, stirring constantly until melted and smooth. Sieve into a small bowl to remove any large pieces of fruit.
Now slice the cooled cake in half with a long serrated knife like a bread knife. Lift off the top half and spread 1 tablespoon of the melted jam all over the lower half which will moisten the surface. Now spread the remaining 1/4 cup of thick, non-melted jam over this.
Cover with the top half of the cake, press down gently and then spread the remaining melted jam all over the top and sides using a blunt knife or small spatula, making sure it's a very thin layer all around otherwise it won't dry. If you have any melted jam leftover, that's ok, just refrigerate it to use on toast the next day!
Let the cake rest at room temperature for about 2 hours, or until the jam on the top and sides feels dry to the touch. This is important so that when you pour the warm chocolate glaze, the jam doesn't melt and mix with the chocolate.
Place the cake on a wire rack with a tray underneath before you make the glaze.
Make the chocolate glaze (see notes for an alternate, easier glaze). Add the sugar and water in a small saucepan on low heat. Stir constantly and bring to a boil. Let the syrup boil for 4 to 5 minutes then pour into a heat-proof bowl. Let it cool for a couple of minutes until still warm, but not boiling hot. In the meantime, add some boiling water to the empty saucepan.
Add the chocolate gradually to the syrup stirring as you go. Be gentle and once all the chocolate has melted, stop stirring. The glaze should coat the back of the spoon. If it seems too thick and not spreadable, add a couple of spoons of the hot water from the saucepan (which will also now have some residual sugar in it) to help thin it out.
Pour the glaze immediately over the top of the cake in a circular motion, letting it drip over the sides. Working as quickly as you can and with very few strokes, use a long metal spatula to spread the glaze in an even layer over the top and sides. This glaze sets almost instantly so don't spread it too much or it will look more gritty than smooth. My cake wasn't the most refined looking and it takes some practice so don't worry, it's going to taste great either way :)
Let the cake rest at room temperature. The glaze will darken and set and feel firm to the touch in about 2 hours. It will have a crackly yet soft consistency once sliced. Cut into 8 pieces and serve with whipped cream if you like! Sacher Torte tastes better the next day, but I do think the glaze consistency is best on day one.
Ideally, store the cake at room temperature in an airtight tin (3 to 4 days). However, in humid weather, it is better to store it in the fridge (about a week) and then let individual slices soften a little before eating. This will also soften the glaze and it won't be crackly anymore but still delicious of course. Happy baking!