make the joconde
Divide the oven into thirds by positioning a rack in the upper third of the oven and the lower third of the oven. Preheat the oven to 425 degrees (F).
Line two 12-inch x 15-inch (31 x 39-cm) jelly-roll pans with parchment paper and brush with melted butter.
In the bowl of a stand mixer fitted with the whisk attachment beat the egg whites until they form soft peaks. Add the granulated sugar and beat until the peaks are stiff and glossy. If you do not have another mixer bowl, gently scrape the meringue into another bowl and set aside.
If you only have one bowl, wash it after removing the egg whites or if you have a second bowl, use that one. Attach the paddle attachment to the stand mixer and beat the almonds, powdered sugar and eggs on medium speed until light and voluminous, about 3 minutes.
Add the flour and beat on low speed until the flour is just combined (be very careful not to overmix here!!!).
Using a rubber spatula, gently fold the meringue into the almond mixture and then fold in the melted butter. Divide the batter between the pans and spread it evenly to cover the entire surface of each pan.
Bake the cake layers until they are lightly browned and just springy to the touch. This could take anywhere from 5 to 9 minutes depending on your oven.
Put the pans on a heatproof counter and run a sharp knife along the edges of the cake to loosen it from the pan. Cover each with a sheet of parchment or wax paper, turn the pans over, and unmold. Carefully peel away the parchment, then turn the parchment over and use it to cover the cakes. Let the cakes cool to room temperature.
make the ginger turmeric syrup
Stir all the syrup ingredients together in the saucepan and bring to a boil. Remove from the heat and let cool to room temperature. Strain through a fine mesh sieve, pressing on the solids.
make the peppermint butter cream
Combine the sugar, and water in a small saucepan and warm over medium heat without stirring, until the syrup reaches 250 degrees F on a candy or instant-read thermometer then remove from heat.
While the syrup is heating, begin whisking the egg and egg yolk at high speed in the bowl of your mixer using the whisk attachment until pale and foamy.
When the sugar syrup reaches the correct temperature, reduce the mixer speed to low and begin slowly (very slowly) pouring the syrup down the side of the bowl being very careful not to splatter the syrup into the path of the whisk attachment. Some of the syrup will spin onto the sides of the bowl but don't worry about this and don't try to stir it into the mixture as it will harden!
Raise the speed to medium-high and continue beating until the eggs are thick and satiny and the mixture is cool to the touch (about 5 minutes or so).
With the mixer on medium speed, begin adding in a few chunks of butter at a time. When all the butter has been incorporated, raise the mixer speed to high and beat until the butter cream is thick and shiny. Finish with about 8 drops of the peppermint oil.
Refrigerate the butter cream until it's firm but still spreadable. Put it back on the mixer and whisk until fluffy.
make the white chocolate mousse
Melt the white chocolate and the 3 Tbs of heavy cream in a small saucepan over low heat.
Stir with a silicon spatula to melt evenly and make sure it doesn't burn. When the chocolate is mostly melted, take it off the heat, add the vanilla and stir until it's all smooth and there are no lumps of chocolate left. Allow it to cool.
Whip the remaining 1 C of heavy cream until soft peaks form.
Add the cooled chocolate to the cream and beat until you have nice firm peaks (careful not to overbeat). If it's too loose, put it in the fridge for a bit to set up.
assembling the cake
Cut a 10" x 10" square and a 10" x 5" rectangle from the center of each sheet of cake. You should end up with 4 pieces (2 of each size).
Place one square of cake on a prep surface and moisten it gently with the flavoured syrup. Spread about half of the buttercream over this layer.
Top with the two rectangular pieces of cake, placing them side by side to form a square. Moisten these pieces with the flavoured syrup. Spread the remaining butter cream on the cake.
Top with the third square of joconde. Use the remaining syrup to wet the joconde and then refrigerate until very firm (at least half an hour).
Prepare the mousse (if you haven't already) and then spread it on the top of the last layer of the joconde. I like to use a pastry knife to make sure my corners are square and that I have a very flat surface on top. Refrigerate for at least two to three hours to give the mousse the opportunity to firm up.
make the glaze
Melt the white chocolate with the heavy cream. Whisk the mixture gently until smooth. Let cool for 10 minutes and then pour over the chilled cake. Spread out into a thin layer allowing it to flow over the edge of the cake. Refrigerate the cake again to set the glaze.
Serve the cake slightly chilled.