Whisk together the egg, oil, both sugars, and vanilla extract in a bowl until smooth. Grate in the apple and then add the flour, baking powder, cinnamon, salt, and allspice; mix with a rubber spatula until evenly combined.
Coat a 6-inch fluted tube pan with Wilton Bake-Easy Non-Stick Spray and transfer the cake batter into the pan. Bake until a cake tester or toothpick inserted in the thickest part of the cake comes out clean or with just a few crumbs, about 45 minutes. Transfer to a wire rack to cool for at least 20 minutes in the pan before turning out the cake. Serve.