Preheat the oven to 475 degrees. Lightly grease a 9″ springform pan (or a 9″ deep dish pie plate may be used).
To make the crust, cream together the butter, 1/3 cup sugar and the one teaspoon vanilla in a bowl. Add the flour and beat to mix well – the mixture will resemble coarse crumbs. Evenly spread the mixture in the pan and press down to completely cover the bottom.
To prepare the filling, beat together the cream cheese, ¼ cup of sugar, egg and ½ teaspoon vanilla until smooth. Pour the filling over the uncooked crust and gently spread until it evenly covers the crust.
In a medium bowl, toss the peeled and sliced apples with the remaining ¼ cup of sugar and Apple Pie Spice. Arrange the apples over the cream cheese layer in a pinwheel design, starting from the outside edge and working toward the middle of the pan.
Any extra apples can be piled on top of the others. Sprinkle the top evenly with the almonds.
Bake at 475 for 10 minutes. Lower the heat to 375 degrees and bake for 25 minutes. Finally, lower the heat to 350 degrees and bake for 20 minutes more.