Grease a 9x13" pan and preheat oven to 350 degrees.
Sift together the flour, baking soda, baking powder, salt and spices.
In a large bowl, use an electric beater to beat the butter until creamy. Add the sugar and beat until light and fluffy, about 3 minutes. Add the egg and beat until smooth.
Add the flour mixture to the mixer bowl in three parts, alternating with the applesauce, beginning and ending with the flour mixture. Scrape down the bowl, then mix on low speed briefly until incorporated.
Pour batter into prepared pan. Bake for 55-60 minutes until the surface springs back when you poke it with your finger or a toothpick inserted in the center of the cake comes out clean. Let cake cool before removing from pan.
Optional: once cool, cut out 2 6" cake rounds using a 6" cake pan as a guide. Wrap tightly in plastic and freeze up to a week. Line the 6" cake pan with plastic wrap and place the remainder of the scraps in an even layer across the bottom of the pan to form a "third" cake layer. Fold the plastic wrap over the cake and freeze until use, up to one week.