Makes one 8-by-5-inch loaf
1. Preheat the oven to 350°F.
2. Grease and lightly flour an 8-by-5-inch loaf pan to keep the cake from sticking.
3. In a small bowl, whisk together the flour, baking soda, baking powder, cinnamon, salt, and allspice.
4. In the bowl of an electric mixer fitted with the paddle or beater attachment, cream together the butter and sugar on medium speed, scraping down the bottom of the bowl at least once, until the mixture is smooth and light in color, about 3 minutes.
5. Add the egg and vanilla extract and mix until the mixture is smooth, scraping down the side of the bowl as needed to ensure everything is fully incorporated, about 1 minute.
6. Add half of the flour mixture and mix just until combined, scraping down the bowl as needed. Add half of the applesauce, and mix until combined.
7. Add the remaining flour and the remaining applesauce, mixing until combined after each. Mix on low speed until the ingredients are well incorporated and the batter is smooth, about 3 minutes.
8. Pour the batter into the prepared loaf pan and bake until a toothpick inserted into the center of the cake comes out clean, about 45 minutes.
9. Remove the cake from the oven and allow to cool slightly in the pan. Run a knife between the cake and the sides of the pan to loosen, then flip it over to remove the cake and place onto a cooling rack. Let cool completely before serving.