Preheat the oven to 325°F.

Butter three 8” or 9” cake pans and line the bottom of each pan with a piece of round parchment paper cut to fit.

In a medium mixing bowl, sift together flour and baking powder. Add the kosher salt, and mix with a whisk to distribute evenly. Set aside.

Add egg, sugar and vanilla bean paste to the bowl of a stand mixer and beat until light and creamy, about 2 minutes. Add the flour mixture and stir until just combined.

Microwave the milk in a large heatproof measuring cup or bowl for 1 minute and 50 seconds, add butter to the hot milk and stir to melt completely. Then add the oil to the hot milk and stir. Add the hot milk mixture to the stand mixer bowl while stirring to combine. Scrape the bowl with a spatula to ensure thorough mixing.

Divide cake batter evenly between the 3 prepared cake pans, about 580 g of batter per pan.

Bake in the middle oven rack for 35 - 38 minutes until a toothpick inserted in the middle of the cake comes out clean.

Remove from the oven and allow to cool to the touch. Run a butter knife along the edge to loosen the cakes if necessary and remove them from the pan. Peel off parchment paper at the bottom and allow to cool completely on a wire rack.

Toss the diced apple, apple cider, brown sugar, apple pie spice and salt in a medium mixing bowl.

Brown the butter in a medium size saucepan over medium heat, about 1.5 - 2 minutes, immediately add the apple mixture to the saucepan. Cook until the sugar dissolves completely and the apple pieces are just starting to soften, about 4 minutes.

Mix the cornstarch with cold water, and add this cornstarch slurry into the saucepan, stir until the pie filling thickens, about 1 minutes. Take off the heat and set aside to cool completely.

Add softened butter, cream cheese, and vanilla extract to a stand mixer bowl and beat on medium speed with the whisk attachment just until creamy.

Add powdered sugar and cinnamon, stir to mix, then slowly increase mixer speed to medium and beat just until all the sugar is incorporated and the frosting is creamy and fluffy.

Add 1 cup of frosting to a pastry bag fitted with a large round tip (size 807) and set aside.

Line a cake stand or plate with a few strips of parchment paper and place 1 layer of cake on top.

Add ¾ cup of frosting on top of the cake and spread it out evenly with a spatula. Pipe a border around the edge of the cake layer using the previously prepared frosting in a pastry bag. Add half the apple pie filling within the frosting border and place the second layer of cake on top. Repeat this step for the second layer.

Add ½ cup of frosting on top of the third layer of cake and use the remaining frosting to frost the side of the cake.

Decorate the cake with apple slices and cookie crumbs if desired. Refrigerate for 2 hours to let the frosting set before serving.