Preheat the oven to 350℉ (177℃). Line three 8-inch cake pans with parchment paper circles and spray with baking spray or brush with cake release.
In a small saucepan, brown 4 tablespoons of the butter over medium heat. It will melt, then foam, then you will start to know small brown flecks on the bottom of the pan and a nutty aroma, all within about 5 minutes. Watch carefully and stir frequently so it doesn't burn. Remove from heat once you see the brown butter flecks in the bottom of the pan. Let cool to room temperature.
10 Tablespoons salted butter
For the cake: In a large mixing bowl, combine the remaining 8 tablespoons of softened butter, the cooled browned butter, and both types of sugar. Beat for 2-3 minutes on medium-high speed until light using a large mixer fitted with the paddle attachment or electric hand mixer. Add the eggs and vanilla. Beat again for 2-3 minutes, stopping to scrape the bottom and sides of the bowl until smooth and combined.
1 3/4 cups (350g) granulated sugar, 1/2 cup (100g) light brown sugar, 4 large eggs, 2 teaspoons pure vanilla extract
With the mixer running on low speed, slowly add the buttermilk and oil. Once both have been added, increase the speed to medium-high and mix for 5 minutes until almost doubled in volume and very airy and light.
3/4 cups (180ml) buttermilk, 1/2 cup (118ml) vegetable oil or canola oil
Add the flour, baking powder, and salt, mixing on low speed until just combined and no dry streaks remain. Divided the batter evenly between the prepared pans and bake for 22-26 minutes until golden brown on top and a toothpick inserted into the cakes comes out clean with just a few crumbs clinging to it.
2 1/4 cups (288g) cake flour, 1 1/2 teaspoons baking powder, 1 teaspoon salt
For the pie crumbs: Combine the flour, sugar, and salt in a large bowl. Add in the butter and water, stirring by hand with a fork until the mixture starts to clump together in small clusters. Sprinkle these in an even layer onto a baking sheet lined with parchment paper. Bake at 350℉ (177℃) for 25 minutes, stirring once halfway through, until golden brown but still moist. The crumbs will continue to firm up as they cool.
2 1/2 cups (353g) all-purpose flour, 3 Tablespoons granulated sugar, 1 teaspoon salt, 1/2 cup (113g) salted butter, 2 Tablespoons water
For the cheesecake filling: Reduce the oven temperature to 300°F. In another large bowl, beat the cream cheese and sugar until creamy and smooth, about 2 minutes. Add the cornstarch, salt, milk, and eggs. Mix again on medium-high speed until completely smooth, about 2-3 minutes, stopping to scrape the bottom and sides of the bowl. Pour into a 9x5-inch loaf pan and spread in an layer. Bake for 20-24 minutes until just set around the edges but still jiggly in the center. Do not overbake. Cool completely at room temperature, then refrigerate until ready to use.
12 ounces full-fat cream cheese, 1 1/4 cups (250g) granulated sugar, 1 1/2 Tablespoons cornstarch, 1/2 teaspoon salt, 3 Tablespoons milk, 2 large eggs
For the apple pie filling: Peel, core, and medium-dice the apples. You can transfer them to a bowl of cold water sprinkled with 1 teaspoon of salt to soak while you work to prevent them from turning brown, if desired, and drain just before using. Add the prepared apples to a large skillet or saucepan with the butter, brown sugar, cornstarch, cinnamon, and salt. Bring to a gentle boil over medium heat, stirring gently until the apples start to release their juices and the brown sugar has melted. Decrease the heat to medium-low and let the apples simmer for about 5 minutes until softened but still retaining their texture and shape and the liquid has started to thicken. Set aside to cool completely. Refrigerate until ready to use.
4 medium Granny Smith apples, 2 Tablespoons salted butter, 1 1/4 cups (250g) light brown sugar, 2 Tablespoons cornstarch, 1 teaspoon ground cinnamon, 1/4 teaspoon salt
For the frosting: Combine about 1/3 of the pie crumbs from above with the milk and salt in a large blender or food processor. Blend until smooth. In another bowl, combine the butter and powdered sugar. Beat for 2-3 minutes with the whisk attachment or an electric hand mixer until smooth and light. Add the pie crumb and milk mixture. Continue to beat for another 2-3 minutes on medium-high speed until light and fluffy.
1/2 recipe pie crumbs, 3/4 cup (180ml) milk, 1/4 teaspoon salt, 5 Tablespoons salted butter, 1/2 cup (60g) powdered sugar
For the cider soak: Whisk cider, brown sugar, and cinnamon in a bowl. Set aside.
1/2 cup apple cider, 2 teaspoons brown sugar, 1/4 teaspoon ground cinnamon
Assembly: To assemble, place the bottom cake layer on a baking sheet lined with parchment paper. Tightly wrap it with an acetate cake collar and secure the collar with tape. Brush or drizzle the cake with about 1/3 of the cider soak mixture, then spread with half of the liquid cheesecake filling. Spoon half of the apple filling onto the cheesecake mixture, then sprinkle with 1/3 of the reserved pie crumbs. Repeat for the next layer of cake with the next cake layer, another 1/3 of the cider soak, remaining cheesecake filling, remaining apple filling, and another 1/3 of the pie crumbs. Top with last cake layer and cider soak, then carefully spread the pie crust frosting onto the cake. Sprinkle with the rest of the pie crumbs for decoration.
Freeze: Transfer the cake to the freezer for 12 hours to set the cake, then thaw for 3 hours on the counter at room temperature before slicing and serving.