Preheat the oven to 325ºF (163ºC). Prepare a 9×13 metal baking dish* by spraying the sides with baking spray and lining the bottom with parchment paper.
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.
Add the batter into your prepared cake pan and smooth the top, then bake for 35-40 minutes, or until a toothpick inserted comes out clean. Let the cake cool in the baking pan until room temperature.