Preheat the oven to 450 degrees F (230 degrees C).
Mix butter, 1/3 cup sugar, and 1 teaspoon vanilla extract in a bowl. Stir flour into mixture to make a workable dough. You might not need all the flour. Press the crust into a 9-inch pie dish.
Mix cream cheese with egg, 1/4 cup sugar, and 1 tablespoon vanilla extract in a bowl. Pour cream cheese mixture into pie crust. Toss sliced apples with 1/3 cup sugar and cinnamon; top the cream cheese filling with the apple mixture. Sprinkle with sliced almonds.
Bake pie in the preheated oven for 10 minutes. Reduce oven temperature to 350 degrees F (175 degrees C) and continue baking until the filling is set, 20 to 25 more minutes. Cool before serving.