Instructions: For the Meringues: Heat the oven to 250 degrees. Line each of 3 baking sheets with parchment paper. Trace an 8-inch circle on the center of each. Set aside.

Using repeated pulses, process the almonds and cocoa in a food processor to mix thoroughly. Mix the powdered sugar and superfine sugar in a bowl; set aside.

Beat the egg whites in a mixing bowl at medium speed until foamy, 1 minute. Add the cream of tartar and then the sugars, 1 tablespoon at a time, until the meringue is shiny and thick, 6 to 7 minutes. Gently fold in the almond mixture. Carefully spoon the meringue into a pastry bag fitted with a plain 1/2-inch tip.

Pipe the meringue in a spiral to completely fill each of the 3 circles. If any meringue is left, change the pastry bag to a star tip and pipe small stars onto the baking sheets between the circles. Theyll be used later for decorating the dacquoise; set aside.

Bake the meringues until they are dry but not browned, 1 hour. Turn off the oven and let the meringues stand in the oven overnight. The meringues can be prepared 2 to 3 days ahead; wrap and store in a dry place.

For the Ganache: Melt the chocolate in the top of a double boiler set over simmering water. Remove the pan from the water; stir in the cream. Whisk until smooth, then blend in the rum. Refrigerate, covered, until firm but not hardened, 1 hour. Remove the ganache to a large mixer bowl and beat until the consistency of frosting, 2 to 3 minutes. Beat the cold cream until it forms soft peaks, 4 to 5 minutes. Stir about 1/4 of the whipped cream into the ganache to lighten, then fold in the remaining cream. Refrigerate at least 1 hour, covered, until ready to use.

For the Buttercream: Combine the sugar and water in a heavy saucepan and heat to boiling. Cook, without stirring, until the mixture reaches 250 degrees on a candy thermometer. Meanwhile, beat the yolks until thick. Pour in the sugar syrup in a thin stream, beating constantly until the mixture cools, about 10 minutes. Beat in the butter, 1 tablespoon at a time. Dissolve the instant coffee in the brewed coffee; beat into the buttercream. Refrigerate, covered, until firm but spreadable, about 1 hour.

For the Whipped Cream: Beat the cream until it starts to hold peaks. Beat in the sugar, instant coffee and rum. Continue beating until stiff, 4 to 5 minutes. Refrigerate 20 minutes.

For Assembly: Place one meringue layer on a platter and spread with a thin layer of buttercream. Cover with ganache, smoothing in an even layer. Top with a second meringue layer and spread with whipped cream. Top with the last meringue layer. Spread the side and top of the dacquoise with the remaining buttercream. Using a pastry bag, pipe any remaining buttercream in a pattern over the top. Sprinkle the top with the almonds and decorate the side with meringue stars. Refrigerate 1 hour.

This recipe yields 12 servings.

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