Mix together the butter, 1/2 teaspoon vanilla extract, all-purpose flour, and 1/2 cup of sugar in a bowl. Press the mixture into the bottom of an ungreased 9-inch springform pan.
Beat together the cream cheese and sugar in a large bowl. Stir in the eggs and 3/4 teaspoons vanilla extract. Pour the egg mixture over the crust.
Combine the sliced apples, cinnamon, and 1/2 cup sugar in a bowl. Spoon apple mixture evenly over the cream cheese mixture. Place the springform pan on a baking sheet.
Bake in preheated oven until center is set, 50 to 65 minutes.
