Special equipment: three 9-inch cake pans; six 9-inch rounds parchment paper; pastry bag with medium round tip
For the cake: Preheat the oven to 375 degrees F. Butter three 9-inch cake pans and line each with a round of parchment paper. Butter the tops of the parchment paper.
Using the whip attachment of a stand mixer, beat the egg whites and salt on high speed until stiff peaks form, about 4 minutes. Set aside.
In a large bowl, use a whisk to beat the egg yolks, granulated sugar and vanilla until thickened and pale yellow. Slowly stir in the flour just until combined. Using a rubber spatula, fold the egg whites into the mixture just until blended.
Pour one-sixth of the batter into each cake pan. Bake until the tops are golden brown, about 10 minutes. Turn the cakes out of the pans, remove the parchment paper, turn them right-side up and cool completely on a rack.
Wash the pans and repeat the process with the remaining butter, parchment and batter.
For the filling: Put the chocolate, espresso powder and vanilla in a medium heatproof bowl set over, not in, a saucepan of simmering water. Stir occasionally until the chocolate is melted. Remove the bowl from the heat and cool slightly.
Beat the butter and confectioners¿ sugar in a stand mixer fitted with a paddle attachment until fluffy. Add the chocolate mixture and beat until blended. Beat in the eggs one at a time, then beat until light and fluffy.
Make the glaze: Put 1 of the cooled cake layers on a large plate. Butter a large, sharp knife.
Put the granulated sugar in a small saucepan over low heat and cook until it begins to turn amber brown. Pour the hot caramel over the cake layer and carefully tilt the plate so that the caramel coats the cake evenly. Working quickly so that the sugar doesn't set, cut the layer into 12 to 16 wedges using the buttered knife. Separate the wedges after they cool a few minutes and cool completely.
To assemble the torte: Put about 1/2 cup of filling into a pastry bag fitted with a medium round tip. Set aside another 1 cup in a small bowl. Spread the remaining filling over the tops of the 5 unglazed cake layers.
Stack the 5 frosted layers on top of each other on a serving platter. Place the glazed cake wedges on top so that they form 1 single layer on the cake.
Frost the sides of the cake with of the reserved 1 cup filling. Pipe the filling in the pastry bag decoratively around the bottom and top edges of the cake. Let the cake stand in a cool place for 30 minutes or so to set the frosting.
To serve, slice between the glazed wedges.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.
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