Line a 34x24 cm pan with parchment paper and preheat the oven to 180°C. Whisk 2 egg whites with sugar until medium peaks form, then fold into a mixture of ground almonds, powdered sugar, and cake flour.
Gently fold melted butter into the batter, pour it into the pan, smooth it out, and tap to remove air bubbles. Bake for 10–12 minutes until golden, then cool completely on a wire rack.
Boil 100g espresso with 25g sugar, stirring until dissolved. Add 1 tbsp of rum, mix well, and let it cool.
Melt 80g dark chocolate with 60g heavy cream over a double boiler. Stir in 20g soft butter, cool, and transfer to a piping bag.
Cook a mixture of egg yolks, sugar, espresso, and milk to 80°C, then strain and cool. Whisk with 120g softened butter in 3 batches using an electric mixer with a whisk attachment, mix in dissolved coffee, and transfer to a piping bag.
Trim the edges of the cooled Biscuit Joconde and cut it into three equal rectangles.
For the first layer, spread a thin coat of chocolate ganache, chill for 15 minutes, then flip it chocolate-side down. Brush with coffee syrup and pipe a layer of coffee buttercream.
For the second layer, brush both sides with coffee syrup, place it on top, and spread chocolate ganache over it.
For the third layer, brush both sides with coffee syrup, place it on top, and finish with a thick layer of coffee buttercream. Smooth and refrigerate for at least an hour to firm up.
Melt 80g semisweet chocolate with 15g oil and cool to 28–30°C. Pour over the cake, smooth the surface, trim the edges, and refrigerate until set.
Cut the cake into slices, savoring the cake layers of almond sponge, coffee buttercream, ganache, and glaze. Serve and delight in every bite!