Preheat oven to 350°F (175°C) and grease and flour three 8-inch round cake pans.

In a large bowl, whisk together cake flour, sugar, baking powder, and salt.

Add 1 cup softened butter to the dry ingredients and mix on low speed until the texture resembles coarse sand.

In a separate container, whisk together milk, egg whites, vanilla extract, and 1 teaspoon peppermint extract.

Slowly pour the liquid mixture into the flour mixture while beating on medium speed until fully combined and aerated, approximately 2 minutes.

Divide the batter evenly among the three prepared pans.

Bake for 30 to 35 minutes, or until a toothpick inserted into the center comes out clean.

Allow cakes to cool in pans for 10 minutes before transferring to wire racks to cool completely.

To make the ganache, heat 1/4 cup heavy cream until simmering, then pour over the chopped white chocolate; let sit for 5 minutes and stir until smooth.

Prepare the frosting by beating 2 cups softened butter with powdered sugar, 2 tablespoons heavy cream, and 1/2 teaspoon peppermint extract until light and fluffy.

Place the first cake layer on a serving plate, spread with half of the white chocolate ganache, and then a layer of buttercream.

Repeat with the second layer, then top with the final cake layer.

Frost the top and sides of the cake with the remaining peppermint buttercream and decorate with crushed candy canes.