Preheat oven to 350°F (175°C) and butter three 7-inch (18-cm) cake pans. Dust all three with flour, shake out excess and set aside.
Sift flour, baking powder and spices in a medium bowl and set aside. Cream butter in another bowl, add the sugars and beat until fluffy, about 5 minutes. Beat in vanilla and salt, then eggs one at a time until incorporated.
Whisk 1/2 of the flour mixture into the wet batter just until incorporated. Next, whisk in apple cider and sauce. Once incorporated add the rest of the flour and whisk until no dry patches are left.
Divide batter evenly between the pans and bake for 20-30 minutes, until a toothpick inserted in the center of each cake comes out clean. Cool cakes in the pans for about 10 minutes, then flip them upside down on a clean kitchen towel and let cool completely before assembly. (At this point you can wrap each cake and store all three in the refrigerator for a few hours or overnight, as that makes the cake easier to assemble.)