Preheat oven to 350 degrees. Butter a 10-inch round cake pan, and line bottom with parchment paper. Butter, and set aside. Spread whole almonds in a single layer on a rimmed baking sheet; toast in oven, tossing occasionally, until light golden and fragrant, 7 to 10 minutes. Let cool.
Place the toasted almonds and 3/4 cup sugar in a food processor, and pulse until finely ground. Transfer to a bowl, and stir in flour; set aside. Melt butter in a small saucepan; remove from heat, and stir in rum and vanilla. Let cool slightly.
In the bowl of an electric mixer, whisk together the egg whites and salt until soft peaks form. Gradually beat in the remaining 3/4 cup sugar; continue beating just until stiff peaks form.
Fold flour mixture into the egg-white mixture in three batches, alternating with butter mixture, beginning and ending with flour mixture. Pour into prepared pan; smooth the top, and bake until cake is golden brown and firm to the touch, about 40 minutes. Let cake cool in pan 20 minutes on a wire rack. Turn out cake onto wire rack, and let cool top side up.
Using a 3-inch oval cookie cutter, cut out eight shapes from cake. Discard the scraps. Trim each to resemble a pinecone, with a rounded base and a pointed tip. Leaving bottom unfrosted, spread about 1/3 cup frosting over each; smooth and round to create a three-dimensional shape.
Starting at the base of one cake, arrange sliced almonds starting in a spiral, overlapping slightly. Continue arranging almonds in closely set rows to cover the entire surface. Transfer finished cake to a baking sheet in the refrigerator to set while repeating with remaining cakes. Just before serving, dust the cakes with confectioners' sugar and garnish with chocolate shavings, if desired.
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