Begin by preheating the oven to 350 degrees Fahrenheit (175 degrees Celsius). Next, lightly grease a 9-inch springform pan and place a parchment paper circle at the bottom. Make sure to grease the paper for easy removal.
To prepare the cake, start by melting chocolate in a double boiler. Stir consistently to prevent scorching, and continue until the chocolate is completely melted, which should take approximately 3 to 5 minutes. Afterward, remove it from the heat and allow it to cool for 5 to 10 minutes.
Using an electric mixer, blend butter and 1/4 cup plus 2 teaspoons of confectioners' sugar in a large bowl until it reaches a creamy consistency. Afterward, incorporate the melted chocolate, and then gradually beat in the egg yolks, one at a time.
In a glass bowl, whip egg whites until they're foamy. Gradually add 1/2 cup plus 2 tablespoons of white sugar while continuing to beat until you achieve stiff, glossy peaks.
Fold egg whites into the chocolate mixture, followed by the cake flour. Pour the resulting batter into the prepared pan and ensure a smooth top.
Bake in the preheated oven for approximately 45 minutes or until the edges start to separate from the pan and a toothpick inserted into the center emerges dry.
After cooling on a wire rack for 15 minutes, gently run a knife along the edge to remove the pan sides. Allow the cake to cool completely on the pan base for an additional 15-20 minutes.
As the cake cools, let's whip up the filling: In a small saucepan, bring 1/4 cup of water and sugar to a boil. Stir until the sugar dissolves, and the syrup becomes clear, which should take about 2-3 minutes. Afterward, remove it from the heat and blend in 2 tablespoons of rum.
Smoothly blend apricot preserves with 1 tbsp water in a blender. Transfer the mixture to a saucepan and bring it to a simmer over medium heat. Cook for about 2 minutes until it thickens, then stir in the remaining 1 tbsp of rum.
After cooling, remove the pan base and parchment paper from the cake. Slice it horizontally into two layers. Begin by placing one layer on a wire rack. Brush it with 1/3 of the syrup and spread 1/3 of the apricot purée. Then, add the second cake layer. Brush the top and sides with the remaining syrup and generously spread the rest of the purée. Let the cake chill on the wire rack until the icing is prepared.
To prepare the icing, melt chocolate in a double boiler over simmering water, ensuring you stir regularly to prevent scorching. This process should take approximately 3 to 5 minutes. Once melted, remove from heat and allow it to cool briefly.
As the chocolate melts, simultaneously simmer the cream in a small saucepan. Stir the cream into the melted chocolate until thoroughly combined. Allow the mixture to cool slightly, stirring regularly, until the icing achieves a spreadable consistency, which should take about 5 to 10 minutes.
Retrieve the cake from the refrigerator, then position a wire rack over waxed paper to catch drips. Gently pour the icing over the cake, ensuring it covers both the top and sides. Allow any excess icing to drip through the rack, and patiently wait until the icing reaches room temperature.
With care, transfer the cake to a serving platter, and then serve it at room temperature.