For the White Cake Layers

• 2½ cups cake flour
• ¾ cup all-purpose flour
• 1 tablespoon baking powder
• 1 teaspoon baking soda
• ¾ teaspoon salt
• ½ cup (1 stick) unsalted butter, softened
• ½ cup vegetable shortening
• 1¾ cups sugar
• 1 tablespoon pure vanilla extract
• 1 large egg
• 1½ cups ice cold water
• 3 large egg whites, at room temperature
• ¼ teaspoon cream of tartar

For the White Chocolate Frosting

• 6 ounces white chocolate, coarsely chopped
• 1½ cups sugar
• ⅓ cup all-purpose flour
• 1½ cups milk
• ⅓ cup heavy cream
• 1½ cups (3 sticks) unsalted butter, soft but cool, cut into small pieces
• 1 teaspoon pure vanilla extract

Make the White Cake Layers

Preheat the oven to 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 into 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 center of each 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.