How to Make It
In a medium saucepan, combine the sugar with the water and lemon juice and bring to a boil over moderately high heat. Using a wet pastry brush, wipe down the side of the saucepan. Simmer the syrup undisturbed until it begins to brown around the edges, 4 to 5 minutes. Swirl the pan, then simmer until the caramel turns a medium-dark amber, about 3 minutes longer. Remove from the heat. Carefully add 1/2 cup of the cream in a steady stream. Stir with a long-handled wooden spoon to combine, then add the remaining 1 1/2 cups of cream. Stir in the vanilla. Transfer the caramel to a bowl and refrigerate until very cold, at least 3 hours or overnight.
Preheat the oven to 350°. Butter the bottoms only of three 9-inch round cake pans and line them with parchment paper. In a medium bowl, whisk the cocoa with the boiling water until a paste forms. Whisk in the milk until smooth; let cool.
Onto a sheet of wax paper, sift the cake flour with the baking soda , baking powder and salt. In a large bowl, using an electric mixer, beat the butter with the sugar at medium speed until light and fluffy. Add the egg and egg yolk and beat until creamy. Add the dry ingredients in 2 batches, alternating with the cocoa mixture; beat until smooth.
Divide the batter evenly among the 3 pans (the layers will be shallow) and smooth the tops. Bake the cakes for 12 to 14 minutes, or until the tops spring back when lightly pressed. Let the cakes cool in the pans for 15 minutes, then run a knife around the edges. Turn them out onto a wire rack to cool completely. Peel off the parchment.
In a small microwave-safe bowl, dissolve the instant coffee in 3 tablespoons of cold water. Sprinkle on the gelatin and let stand until softened, about 5 minutes. Using an electric mixer, beat the caramel with the crème fraîche just until soft peaks form. Working quickly, microwave the coffee-gelatin mixture at 10-second intervals just until the gelatin is completely dissolved and hot to the touch. Quickly whisk the hot gelatin into the caramel.
Line a 9-inch round springform pan with plastic wrap, leaving at least 4-inches of overhang all around. Place 1 of the cake layers, flat side down, in the bottom of the springform pan. Spoon half of the caramel mousse over the cake in the pan and top with another cake layer. Press lightly. Add the remaining caramel mousse and top with the final cake layer, flat side up. Fold the overhanging plastic wrap over the cake and refrigerate for at least 3 hours or overnight.
In a medium saucepan, heat the cream just until small bubbles appear around the edges. Remove from the heat. Add the chocolate and let stand for 5 minutes. Stir until smooth. Let the glaze cool for about 1 hour, until thick but pourable. In a small bowl, stir the confectioners' sugar with the milk and transfer to a pastry bag fitted with a tiny, plain tip.
Unwrap the top of the cake and invert the cake onto a large serving plate. Slide wide strips of wax paper under the bottom edge of the cake all around to catch drips. Remove the side and bottom of the pan and unwrap the cake. Smooth the sides of the cake with an offset spatula. Pour about 3/4 cup of the chocolate glaze over the cake and spread it all around, filling in any gaps. Refrigerate for about 5 minutes, until set. Pour the remaining chocolate glaze on the cake and spread it all around; do not spread once the glaze is smooth.
Starting from the center and working quickly out toward the edge, pipe a continuous spiral of white glaze over the top of the cake. Using a toothpick or wooden skewer, run the tip gently through the glaze from the center to the edge to create a web pattern. Refrigerate the cake for about 15 minutes to set the glaze. Let the cake stand at room temperature for 30 minutes. Gently remove the wax-paper strips, cut the cake into wedges and serve.