Heat the oven to 350°F (175°C). Grease an 8- or 9-inch round cake pan and line the bottom with parchment paper.

Beat the room-temperature eggs, sugar, and salt for 8–10 minutes, until very pale, thick, and tripled in volume. Sift in the flour in two or three additions, folding gently after each addition, then fold in the vanilla.

Bake the sponge for 25–35 minutes, until the center springs back and a toothpick comes out clean. Cool completely, then wrap and refrigerate for several hours or overnight for easier slicing.

Whisk the egg yolks, sugar, flour or cornstarch, and salt in a saucepan. Gradually whisk in the milk, cook over medium heat while stirring constantly until thick and bubbling, and continue for about 1 minute; stir in vanilla, cover the surface directly with plastic wrap, and chill.

Divide the cold pastry cream in half. Stir the chopped semisweet chocolate into one portion while warm, gently rewarmed if needed, then chill both fillings until spreadable.

Simmer the water and sugar until the sugar dissolves, then remove from the heat and cool somewhat. Stir in the rum after cooling to preserve more of its aroma, then set aside.

Slice the sponge horizontally into three even layers. Brush the first layer gradually with rum syrup, spread with vanilla pastry cream, add the second layer, moisten it, spread with chocolate pastry cream, then add the final layer and moisten it lightly.

Whip the cold cream with powdered sugar and vanilla to medium-stiff peaks. Frost the chilled cake on top and around the sides, press toasted sliced almonds onto the sides if desired, and refrigerate for at least 4 hours before serving.