Preheat the oven to 350ºF and prepare three 6-inch or two 8-inch or 9-inch cake pans by spraying the sides with cooking spray and fitting the bottoms with a wax or parchment paper circle.
In a medium bowl, add the flour, baking soda, baking powder, salt, cinnamon, nutmeg, and allspice. Whisk together and set aside.
In the bowl of a stand mixer fitted with the paddle attachment (or with a hand mixer and a large bowl), beat the butter on high speed until light and creamy, about 2 minutes. Add the brown and white sugars and continue to beat on medium-high until light and fluffy, 2-3 minutes. Scrape down the bowl and paddle as needed.
Turn the mixer to low speed and add the vanilla and the egg. Turn the mixer to medium and beat for one full minute.
Turn the mixer back to low speed and add the flour mixture in three parts, alternating with the applesauce, beginning and ending with the flour mixture. When it begins to come together after the last flour addition, turn off the mixer and give it a few stirs by hand to make sure everything is incorporated. The batter will be very thick.
Divide the batter between your prepared cake pans (fill no more than 2/3 full) and smooth the tops, then bake for 36-42 minutes, or until a toothpick inserted comes out clean. Let the cakes cool completely before filling and frosting.