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Preheat the oven to 350F (180C).
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Line two 11-by-17-inch (28-by-43-cm) rimmed baking sheets with parchment paper.
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To make the cake base, spread the hazelnut powder on another rimmed baking sheet, place in the oven, and bake for about 5 minutes, or until it smells like roasting nuts.
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Remove from the oven and pour into a bowl to cool.
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In a stand mixer fitted with the whip attachment (or by hand with determination), whip together the eggs and 1 2/3 cups (180 g) of the powdered sugar until white and creamy.
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Using a rubber spatula, fold in the hazelnut powder and flour just until thoroughly combined.
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Transfer the mixture to a large bowl, and rinse the whip attachment and the mixer bowl.
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Refit the whip attachment to the mixer, and put the egg whites, the remaining 5 tablespoons (40 g) powdered sugar, and the salt in the bowl.
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Beat until stiff peaks form.
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Gently fold the beaten whites into the egg-hazelnut mixture just until no streaks are visible.
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Divide the batter evenly between the prepared baking sheets, and spread to level with a spatula, preferably an offset spatula.
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Place in the oven and bake for 11 minutes, or until lightly browned and set.
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Let cool in the pans on wire racks for about 10 minutes, then remove the parchment paper by lifting and peeling it slowly off the cakes.
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Let cool completely.
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To make the ganache, put the chocolate in a bowl.
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In a small saucepan, bring the cream to a boil and pour it over the chocolate.
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Let sit for 3 minutes, then whisk until smooth.
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Let the ganache sit for 20 minutes at room temperature to firm up a bit.
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To make the buttercreams, rinse the whip attachment and bowl again, then put the egg yolks in the bowl.
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In a saucepan, combine the sugar and water over medium heat and heat until the sugar melts.
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Clip a candy thermometer to the side of the pan.
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When the sugar mixture reaches 226F (108C), start whisking the egg yolks.
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When it reaches 239F (115C), remove from the heat while slowly pouring in the yolks and whisking constantly.
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Continue whisking until the mixture is at room temperature.
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Add the butter a few pieces at a time to the cooled egg mixture and whisk until well incorporated.
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Scoop out half of the mixture into another bowl.
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Split the vanilla bean lengthwise and, using the tip of a sharp knife, scrape the seeds from the pod halves into one of the bowls and mix well.
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Add the Nutella to the second bowl and mix well.
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If you notice a shiny film forming as you are stirring the buttercreams, quickly add a few drops of cold water and continue stirring.
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The buttercreams are easiest to work with if used right away.
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If youre unable to get your hands on (that is, build) a metal mold in the dimensions described in the headnote, you can use 2 half-gallon (2-liter) milk cartons with the top and one long side cut out of each one.
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Adjust to any desired length by sliding the sides of the two cartons onto one another.
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Tape the bottom of the mold so that it maintains a perfect rectangle, and line the inside of the mold with parchment paper.
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With your mold built, you are ready to build your cake.
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Measure the width of the mold, then cut the cakes into slabs of the same width, minus 1/4 inch (6 mm).
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Cut the slabs into even lengths to fit the mold.
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Place a cake layer, shiny brown side up, in the mold and brush a little of the rum over it.
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Top the cake layer with a layer of the Nutella buttercream, making it slightly thicker than the cake layer.
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Top the buttercream with a second cake layer, shiny brown side up, and brush with more rum.
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Top the cake layer with a layer of ganache, making it slightly thicker than the cake layer.
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Place a third cake layer, shiny brown side up, on the ganache layer and brush with more rum.
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Add a layer of the vanilla buttercream, making it slightly thicker than the cake layer.
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Repeat the layers, finishing with a cake layer topped with a layer of ganache.
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Then add the nuts.
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Let the cake rest in the refrigerator overnight.
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The next day, cut the tape holding the mold together to release the cake.
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You may discover that the whole thing has sunk a bit, or that the cake has soaked the ganache excessively.
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No matter.
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You will have a delicious, silky brick of joy.
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To serve, cut into slices with a hot, wet knife.
