Preheat the oven to 325ºF*. Line 2 12-cup cupcake pans with paper liners.

Sift the cake flour, all-purpose flour, baking powder, baking soda and salt together into a large bowl. Set aside.

In the bowl of a stand mixture fitted with the paddle attachment, beat the butter and the shortening together until creamy, 3 to 4 minutes. Add the sugar and the vanilla and continue to beat on medium for 3 more minutes. Scrape the bowl, then add the egg and beat until combined. Turn the mixture to low and add 1/3 of the flour mixture, followed by half of the ice water, another 1/3 of the flour, the remaining water, then the remaining flour. Scrape down the bowl and mix a few more seconds on low to combine.

In a clean bowl, whisk the egg whites and the cream of tartar until soft peaks form. Gently fold the egg whites into the batter. Fold the coconut into the batter.

Fill the cupcake liners 2/3 full of the batter. Bake until a tester inserted in the center comes out clean, 20 to 25 minutes, rotating the pans half-way through the baking time.

Transfer the pans to a wire rack to cool for 20 minutes, then remove from the pans and allow to cool completely.

Place a fine-mesh sieve over a medium bowl.

Bring the half and half to a simmer in a medium saucepan over low heat. Add the coconut then place a cover on the pan and simmer for 20 minutes to allow the coconut to steep. Strain and discard the coconut.

In a medium bowl, whisk the egg yolks with the sugar, flour and salt until the mixture is pale, about 1 minute. Whisk half of the warm half and half into the egg yolk mixture, then pour that into the saucepan with the remaining half and half. Cook over medium heat, stirring constantly, until thickened, about 6 minutes. Remove the pan from the heat and stir in the vanilla. Strain the cream through the sieve. Press a piece of plastic wrap over the surface of the cream and refrigerate for about 1 hour, or until completely cooled.

In a medium, heavy saucepan, combine the sugar and the flour and whisk together. Add the milk and the cream and cook over medium heat, whisking occasionally, until it comes to a boil and thickens, about 20 minutes.

Transfer the mixture to the bowl of a stand mixer fitted with the paddle attachment. Beat the mixture on high speed until cool. Reduce the speed to low and add the butter, one piece at a time, and continue to mix until thoroughly incorporated. Increase the speed to medium-high and beat until the frosting is light and fluffy.

Add the vanilla and 1/2 cup of the cooled coconut pastry cream. Continue mixing until combined. (If the frosting is too soft, place in the refrigerator to chill slightly, then beat again until it is the right consistency.)

Cut a cone out of each of the cupcakes and fill with about 1 teaspoon of the coconut pastry cream. Replace the top of the cone.

Frost each of the cupcakes with the frosting, then dip the tops into the toasted coconut.