Preheat oven to 450. Cream butter and 1/3 cup sugar until light and fluffy. Blend in vanilla. Add flour; mix well. Spread on bottom and sides of a springform pan. Combine cream cheese and 1/4 cup sugar, mixing until well blended. Blend in egg and vanilla. Pour in pastry lined pan. Toss apples with combined sugar and cinnamon; spoon over cream cheese layer. Sprinkle with nuts. Bake at 450 for 10 minutes. Reduce temperature to 400 and continue baking 25 minutes.