Make the rum syrup: Stir together the warm water and sugar in a small bowl until the sugar is dissolved. Stir in the rum. Let cool and then cover until ready to use.
Split and layer the cake: Trim a very thin layer off the top of the cake with a sharp serrated knife to expose the cake's interior porous structure (you may need to cut as much as 1/4 inch off the top if the cake is slightly domed). Slice the cake in half horizontally with the serrated knife (each layer will be about 3/4-inch-thick).
Dot a bit of pastry cream in the center of a 10-inch cardboard disk (or the removable bottom of a 10-inch tart pan) and set the bottom cake layer, cut side up, on the disk. The pastry cream will hold it in place. (If you have a cake turntable, set the disk on that. Otherwise, put the disk on a large cake plate or cake stand.)
Use a pastry brush to dab half of the rum syrup all over the cake.
Reserve 1/2 cup of the pastry cream and spread the remainder on the top of the cake. The layer of pastry cream will be about 1/2 inch thick. Set the second cake layer over the cream, and dab on the remaining rum syrup. With a narrow metal spatula, spread the reserved pastry cream all around the sides of the cake.
Supporting the cake in your hand, hold it over a sheet tray or a piece of parchment and gently press the almonds all around the sides of the cake. The nuts that don't stick will fall into the tray or onto the parchment. You won't use all the nuts; snack on the leftovers.
Using two large offset spatulas, carefully transfer the cake from the cardboard disk onto a cake plate or cake stand. Pour the warm ganache (if it has started to firm up, warm it gently in a heavy saucepan) onto the center of the top of the cake without letting any run down the sides. Carefully spread the ganache with a large metal icing spatula right to the edges of the cake.
Make the white icing and decorate the cake: In a small bowl, whisk together the confectioner's sugar and hot water. The icing should have a smooth, creamy consistency that's a bit thinner than honey; add a few more drops of water if needed.
Scrape the white icing into a zip-top bag, seal, and cut a tiny hole in one corner of the bag to pipe it. Squeeze the icing over the ganache in a decorative pattern like a crosshatch or zigzag. Refrigerate the cake for at least 1 hour.
Twenty or thirty minutes before serving, remove the assembled cake from the refrigerator. Run a sharp knife under hot water and wipe it dry before slicing. You may have to make several progressively deeper cuts (heating the knife each time) to cut through the chocolate without cracking the top. The rest of the cake cuts easily. Put portions on dessert plates and let stand at room temperature for about 20 minutes before serving. Refrigerate any leftovers for up to two days.
Make Ahead Tips: Make the pastry cream up to one day ahead and refrigerate, covered. The rum syrup and toasted almonds can also be made one day in advance.
French Butter Sponge Cake
Position a rack in the lower third of the oven; heat the oven to 325 degrees F. Melt the butter in a small saucepan over medium-low heat and set it aside to cool. It must be no hotter than tepid when used.
Butter the bottom of a 10-inch springform pan and line it with a round of parchment or waxed paper; butter the paper. Dust the inside of the pan with all-purpose flour; knock out any excess. Sift the cake flour three times onto a sheet of waxed paper, return the flour to the sifter, and set the sifter on the waxed paper.
Using the whip attachment on your mixer, whip the yolks on medium-high speed until thickened, about 3 minutes. Continue beating on medium high and gradually add 1/2 cup of the sugar, about 2 tablespoons at a time; beat for 20 to 30 seconds between additions. You may need to stop to scrape the bowl occasionally. Beat on high speed until very thick and pale, about another 3 minutes. When the whip is raised, the yolks should fall into the bowl and form a slowly dissolving ribbon. Beat in the vanilla. Scrape the mixture into a large, wide bowl. Wash and dry the mixing bowl and the whip.
Combine the egg whites with the salt in the bowl of the mixer and begin mixing on medium low until frothy, about 1 minute. Increase the speed to medium and mix until the whip leaves distinct traces in the whites and forms peaks that curl softly at their tips when the beater is raised, about 4 minutes. With the mixer running on medium, gradually add the remaining 1/2 cup sugar, about 2 tablespoons at a time, beating for 20 to 30 seconds between additions. Continue beating on medium to medium-high speed until the whites hold a firm shape and form peaks that curl only very slightly at their tips, another 1 to 2 minutes. Don't beat until the whites form stiff, unwavering upright peaks.
Check the melted butter to be sure it's tepid (about 95 degrees F) and liquid. The butter must not be at room temperature or it won't blend into the batter properly. If necessary, rewarm it briefly.
Scoop about a quarter of the whites over the yolk mixture and fold together gently with a few broad strokes of a large rubber spatula to lighten the mixture. Rotate the bowl a bit with each fold, and don't be too thorough at this point: It's all right if streaks of white remain. Sift about a third of the cake flour over the mixture and scoop about a third of the remaining whites over the flour. Fold together gently, turning the bowl as you go; but again, don't be too thorough. Sift half of the remaining flour and scoop half of the remaining whites onto the batter and fold in the same manner. Sift on the last of the flour and add the last of the whites; fold gently but thoroughly, only until the batter is an even yellow color.
Drizzle half of the tepid butter on top of the batter and partially incorporate it quickly with three or four broad strokes, turning the bowl a little with each fold. Pour on the remaining butter and fold it in only until no butter shows. Carefully scrape the batter into the prepared pan (the pan will be about half full), spread it evenly, and put the pan in the oven immediately.
Bake until the cake is golden brown and springs back when gently pressed, about 40 minutes; don't overbake. Let the cake cool in the pan on a rack for 10 minutes. Run a sharp knife around the edge to release the cake from the sides; unclasp and remove the bottom of the pan and the paper liner. Replace the paper on the cake bottom and, cover with another rack, and invert. Let cool completely, right side up. If storing, wrap the cake tightly in plastic when completely cool.
Heat the milk in a heavy 3-quart saucepan over medium heat until very hot. If a skin forms on the milk's surface, remove it. Meanwhile, beat the eggs, egg yolks, and salt in a medium bowl with an electric hand mixer on medium-high speed until slightly thickened, about 3 minutes. With the mixer on medium-high speed, add the sugar, 2 to 3 tablespoons at a time, beating about 30 seconds between additions. Continue beating until the mixture is very thick and pale, about another 3 minutes. On low speed, beat in the rum, vanilla, and cornstarch until smooth. Still on low speed, slowly add the hot milk, just a few dribbles at first and then in tablespoon-size dollops, beating after each addition only until smooth.
Scrape the mixture into the saucepan used to heat the milk and add the cold butter pieces. Set the pan over medium heat and stir constantly and gently with a heatproof rubber spatula, scraping all across the bottom and around the sides of the pan. As the mixture heats, it will thicken first at the bottom of the pan, and the mixture will look very lumpy. Decrease the heat slightly and keep stirring as the entire mixture approaches the boiling point and becomes even thicker and lumpier. Switch to a whisk and stir (don't beat) to smooth the pastry cream. The custard won't actually boil because it's so thick. When very thick and smooth (after 7 to 8 minutes), reduce the heat to very low and continue stirring gently with the rubber spatula all over the bottom and sides of the pan for another 1 minute.
Remove the pan from the heat and immediately scrape the custard into a medium bowl. Lay a piece of plastic wrap directly on the custard's surface, let cool to room temperature, and then refrigerate overnight.
In a heavy 1-quart saucepan, bring the cream just to a boil over medium heat. Add the chopped chocolate and stir with a small wire whisk. Take the pan off the heat and stir occasionally as the chocolate melts. The ganache should be perfectly smooth. Let cool slightly before using. If it cools beyond a pourable consistency, warm it gently in a heavy saucepan.