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