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MAKE THE PASTRY: Cream the butter in a food processor.
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Add the confectioners' sugar, ground almonds, egg yolks, cinnamon and salt and process until smooth.
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Add the rum and pulse to blend.
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4
Add the flour and baking powder and pulse to blend.
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Transfer the dough to a floured work surface and gather into a ball.
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Cut the dough into thirds; pat each piece into a 6-inch disk and wrap separately in plastic.
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Refrigerate for at least 2 hours and up to 2 days.
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Let the dough soften slightly before rolling out.
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Preheat oven to 350-degrees.
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Line 3 baking sheets with parchment paper.
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On a lightly floured surface, roll 1 disk of the dough into a 10-inch round.
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Using the bottom of a springform pan as a guide, cut out a 9-inch round.
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Carefully transfer the round to a parchment-lined baking sheet and chill for 30 minutes.
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Repeat with the remaining disks of dough.
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Prick the rounds all over with a fork and then bake them one at a time for 20-25 minutes, or until lighlty browned.
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Transfer the pastry to a rack and let cool completely.
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MAKE THE MOUSSE: In a bowl, combine the apple with 1 Tablespoon of the sugar and the cinnamon.
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In a large heavy skillet, melt the butter over moderately high heat.
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Add the apple and cook over high heat, stirring, until lightly crusted and tender, but not mushy, 3-5 minutes.
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Add the rum, carefully ignite it and stir the apple to coat.
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Transfer to a plate and let cool to room temp.
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In a small saucepan, heat the cream with the cinnamon stick until bubbles appear around the edge.
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Remove from the heat, cover and let steep for 10 minutes.
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Meanwhile, melt the chocolate in a metal bowl set over a saucepan of simmering water.
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Remove from the heat.
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Strain the hot cream over the chocolate and discard the cinnamon stick.
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Gently stir unmtil smooth and let cool until tepid.
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In a bowl, beat the egg whites until very foamy.
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Gradually add the remaining 3 Tablespoons of sugar, beating until stiff glossy peaks form.
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30
Using a large rubber spatula, fold one-quarter of the beaten whites into the chocolate.
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Fold in the remaining whites and then fold in the apple.
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Set 1 cinnamon pastry round in the bottom of a 9-inch springform pan and spread half of the chocolate mousse on top.
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Cover with a second cinnamon pastry round, pressing gently.
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Spread the rest of the mousse over the top and cover with the last pastry round.
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Refrigerate for at least 30 minutes and up to 8 hours.
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GARNISH THE CAKE: Remove the sides of the pan and pat a thin layer of chocolate shavings against the side of the cake.
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Cut 5 thin slices from the apples and rub with lemon juice.
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Fan the apple slices on the cake.
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In a small saucepan, melt the apple jelly over moderately high heat and brush a thin layer over the apples.
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40
Garnish with the cinnamon sticks and serve.