1
For the filling and topping, combine the whipping cream, unsalted butter, and sugar in a heavy saucepan over medium-high heat. Stir until the sugar dissolves and the mixture comes to a simmer. Remove from heat.
2
Add 20 ounces of finely chopped semisweet chocolate to the saucepan, whisking until melted and smooth. Stir in the dark rum and vanilla extract.
3
Transfer 1 cup of the warm chocolate mixture to a small bowl. Whisk in 2 ounces of the remaining chocolate and set aside for the topping, covering it to keep warm at room temperature.
4
Pour the remaining chocolate mixture into a large bowl, cover, and refrigerate until cold and thick, at least 6 hours or overnight.
5
For the syrup, stir together the water and two tablespoons of sugar in a small saucepan over low heat, just until the sugar dissolves. Remove from heat and mix in 2 tablespoons of dark rum. Cover and let it sit at room temperature for up to 1 day.
6
Preheat your oven to 350°F (175°C). Butter a 9-inch square metal baking pan, line the bottom with waxed paper, and butter the paper.
7
In a cup, combine 2 teaspoons of vanilla extract and 3/4 teaspoon of instant coffee powder, swirling them to dissolve.
8
In a small bowl, sift together the cake flour, cocoa powder, and baking soda.
9
Using an electric mixer, beat the egg yolks with 1/2 cup of sugar in a medium bowl until the mixture is thick and light in color, about 3 minutes. Beat in the vanilla-coffee mixture.
10
In a separate large bowl, using clean, dry beaters, beat the egg whites, cream of tartar, and salt until soft peaks form. Gradually add the remaining 1/4 cup of sugar, beating until the whites are stiff but not dry.
11
Gently fold 1/3 of the egg whites into the yolk mixture. Next, fold in half of the flour mixture, followed by half of the remaining egg whites, then the rest of the flour mixture, and finally the remaining egg whites.
12
Transfer the batter to the prepared pan, smoothing it to an even thickness. Bake for about 18 minutes, until the cake is puffed and a tester inserted into the center comes out clean.
13
Allow the cake to cool in the pan on a rack. Once cooled, run a knife around the edges to loosen the cake and turn it out onto a work surface, peeling off the waxed paper. Cut the cake horizontally in half using a long serrated knife.
14
Place one half of the cake, cut side up, on a platter. Drizzle half of the syrup over the cake, about 3 1/2 tablespoons.
15
Using an electric mixer, beat the cold chocolate filling until thick and lighter in color, which should take less than a minute. Spread the filling evenly over the first cake layer.
16
Drizzle the remaining syrup over the cut side of the second cake layer and place it syrup side down atop the filling, gently pressing to adhere.
17
Rewarm the reserved topping over low heat until pourable. Pour it onto the center of the cake and use a small spatula to spread it to the edges, being careful to avoid spills.
18
Chill the cake for up to 1 day, covering loosely with foil or a cake dome after 2 hours, to let the flavors meld. When ready to serve, cut into 16 squares and enjoy.