Preheat oven to 350°F. Grease a 9″x9″ nonstick baking pan with butter.
In a large bowl, whisk together cake flour, cinnamon, baking powder, allspice, nutmeg, and salt.
Using an electric mixer, beat egg, sugar, oil, and applesauce until thick and combined.
Gradually add dry ingredients to wet mixture, beating until smooth.
Mix in apple cider and vanilla until just combined.
Fold in chopped apples, then pour batter into prepared pan.
Bake until a toothpick inserted into the center comes out clean, about 45 minutes.
Let cool for 30 minutes, then cut into squares. Dust with powdered sugar and serve with whipped cream. Enjoy your delicious apple spice cake!