Separate the eggs into yolks and whites. Beat the egg whites with a pinch of salt until they become a glossy, airy foam, then gradually add the sugar while whisking continuously until you get stiff, shiny peaks. In a small bowl, mix the yolks with the oil until smooth, then gently fold this mixture into the whipped egg whites using a spatula.
In another bowl, whisk together the flour and cocoa powder and sift the mixture. Sprinkle it over the egg foam and fold it in with slow, wide movements from bottom to top, just until there are no visible streaks of flour and the batter looks even and airy.
Line a rectangular baking pan (about 9 x 13 inches / 22 x 30 cm) with parchment paper and gently spread the batter in an even layer. Bake in a preheated oven at 340°F (170°C) for about 35–40 minutes, until the top is risen and lightly browned and a toothpick comes out clean. Transfer the baked sponge to a cooling rack and let it cool completely, then slice it horizontally into two equal layers.
For the cream, warm the heavy cream in a small saucepan until it is just about to simmer. Remove from the heat, add the chopped dark chocolate and stir until the chocolate is fully melted and you have a smooth, silky ganache. Let this mixture cool down to room temperature so it thickens slightly.
Beat the softened butter for 2–3 minutes until light and creamy, then gradually mix in the cooled ganache (ideally at 68–77°F / 20–25°C) until you get a smooth chocolate buttercream. For the syrup, melt the sugar in a small saucepan over medium heat until it turns into a light caramel, without burning it. Carefully add the warm water, simmer until the caramelized sugar fully dissolves, then remove from the heat and let it cool. Stir in the rum extract only after the syrup has cooled.
To make the glaze, bring the water and sugar to a boil and cook for 3–4 minutes until it slightly thickens into a light syrup (about 221°F / 105°C). Take the pan off the heat, add the chopped dark chocolate and butter, and stir until everything is melted and the glaze is shiny and smooth.
To assemble, place the first sponge layer back into the pan and generously soak it with part of the rum syrup. Spread the chocolate cream evenly over the entire surface, keeping a small portion aside for decorating the top later. Place the second sponge layer on top and soak it with the remaining syrup. Cut the cake into neat squares and gently separate the pieces so you can glaze each one individually.
Pour or spoon the warm chocolate glaze over each individual piece, letting it run slightly over the sides so they are nicely coated. Pipe small rosettes of the reserved chocolate cream on top of each piece as a classic finishing touch, then chill the cakes for a few hours until the glaze sets and the flavors meld together. These amandine squares are rich, moist and incredibly flavorful.