Make the crust: Cream the butter, sugar, and vanilla in the bowl of a stand mixer until fluffy. Add the flour and mix until the dough comes together in a smooth ball. It will be dry and crumbly at first, but eventually form a soft cohesive mass (like playdough).
Pat the dough into a 9-inch (23cm) springform pan, pushing it halfway up the sides (don't clean out the mixing bowl just yet). Use a flat-bottomed cup or jar to push the dough into the corners and along the sides to smooth them out so the dough is not thicker at the crease around the bottom edge. Level out the top edge all the way around by pushing down on it to make it even. You can scallop the top edge in a pattern by pushing down with a measuring spoon all the way around, if you wish (or just leave it straight).
Then the filling: Into the mixing bowl, add the softened cream cheese (if it wasn't softened enough, put it in the microwave for 20 seconds on high power) and sugar. Cream them together until smooth. Add the egg and vanilla and beat until smooth and lump-free. Pour the cream cheese filling onto the prepared pastry crust and smooth the top.
And finally, the topping: Peel, quarter, and core the apples, then cut out the cores. Cut each quarter into thin lengthwise slices (6 to 8 sliices per quarter). Mix the cinnamon and sugar, then sprinkle it over the apples and toss to coat all the slices.
Arrange the sugared apple slices on top of the cream cheese filling, either in a charming haphazard way or in a pattern of concentric circles. Lay the apple slices out singly (it doesn't take long, I promise) rather than dumping them in, so that they stay on top of the cheesecake layer. Scrape in any apple juices from the bowl.
Sprinkle with the sliced almonds.
Then bake: Bake the apple cake for 10 minutes at 450°F (230°C), then lower the heat to 375°F (190°C) and continue baking for another 25-30 minutes. Remove from the oven and allow the cake to cool in the pan. After 10 minutes, run a knife around the inside of the ring to loosen it, then continue to cool the cake until it is no longer warm. Chill for 1 to 2 hours or longer. Remove the springform ring and put the cake onto a serving plate.
Dust with icing sugar just before serving. Delicious with whipped cream.
Serves 8.
Notes
This crust turns out equally well with my gluten free flour mix as it does with regular all-purpose flour.