Preheat the over the 325 degrees F. Butter three 8-inch round cake pans, line the bottoms with parchment paper, and butter the parchment. Dust with flour, and knock out the excess flour.
Sift the flours, baking powder, baking soda, and salt together in a large bowl. Set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and shortening on medium speed until creamy, 3 to 4 minutes. Add the sugar and vanilla and beat on medium speed until fluffy, about 3 minutes. Scrape down the bowl, add the egg, and beat until just combined. Turn the mixer to low. Add the flour mixture, alternating with the ice water, in three separate additions, beginning and ending with the flour mixture. Scrape down the bowl, then mix on low speed for a few more seconds.
In a medium bowl, whisk the egg whites and cream of tartar until soft peaks form. Do not overbeat. Gently fold the egg whites into the batter.
Divide the batter among the prepared pans and smooth the tops. Bake for 40 to 45 minutes, rotating the pans halfway through the baking time, until a toothpick inserted in the centre of the cake comes out clean. Transfer the cakes to a wire rack and let cool for 20 minutes. Invert the cakes onto the rack, remove the pans, and let cool completely. Remove the parchment.
Make the white chocolate frosting
Using either a double boiler or a microwave oven (see page 23), melt the white chocolate and set it aside to cool.
In a medium heavy-bottom saucepan, whisk the sugar and flour together. Add the milk and cream to cook over medium heat, whisking occasionally, until the mixture comes to a boil and has thickened, about 20 minutes.
Transfer the mixture to the bowl of an electric mixer fitted with the paddle attachment. Beat on high speed until cool. Reduce the speed to low and add the butter; mix until thoroughly incorporated. Increase the speed to medium-high and beat until the frosting is light and fluffy.
Add the vanilla and white chocolate and continue mixing until combined. If the frosting is too soft, put the bowl in the refrigerator to chill slightly, then beat again until it is the proper consistency. If the frosting is too firm, set the bowl over a pot of simmering water and beat with a wooden spoon until it is the proper consistency.
Assemble the cake
Refrigerate the frosting for a few minutes (but no more) until it can hold its shape. Place one cake layer on a serving platter. Trim the top to create a flat surface, and evenly spread about 1 1/4 cups of the frosting on top. Add the next layer, trim and frost it, then add the third layer. Crumb coat the cake and put the cake in the refrigerator for about 15 minutes to firm up the frosting. Frost the sides and top with the remaining frosting. Garnish with a few white sprinkles or white nonpareils and refrigerate for 15 minutes to firm up the finished cake.
This cake will keep beautifully in a cake saver at room temperature (cool and humidity free) for up to 3 days. If your room is not cool, place the in a cake saver and refrigerate for up to 3 days. Remove the cake from the refrigerator and let sit at room temperature for at least 2 hours before serving.