Preheat oven to 350°F. Line three 8” round cake pans with parchment paper and lightly grease. Set aside.
In a mixing bowl sift together the flour, cinnamon, baking soda, nutmeg, and salt.
In another mixing bowl cream together sugars and butter. Add eggs, one at a time, followed by the vanilla.
Pour half the dry ingredients into the wet ingredients and mix together. Add half of the milk and continue to mix the batter together. Repeat with the remaining flour mixture and milk.
Mix until no lumps remain, making sure not to over mix the batter.
Fold in the diced apple until evenly spread throughout the batter.
Evenly divide cake batter into the prepared cake pans and bake for 15-18 minutes or until a toothpick comes out clean when inserted into the center of each cake round.
Allow cake rounds to cool, about 30 minutes to an hour.
For chai spiced buttercream: Place butter into a large mixing bowl and add 1 cup of sugar. Beat until smooth. Add another cup of sugar and continue to beat until smooth. Repeat until all the sugar has been incorporated and buttercream is very thick. Pour vanilla extract and spices into a measuring cup with the milk and whisk together. With the motor running, pour the milk mixture into the buttercream and beat together until fully incorporated and the buttercream is light and fluffy.
To assemble: Using a piping bag fitted with a large star piping tip, fill with buttercream. Pipe uniform stars over the entire surface of one of the cake rounds and top with another cake round. Repeat. Spread a small layer of the buttercream over the top cake round and pipe stars around the perimeter. Flood the center of the cake surface with the shaved white chocolate and serve.