Preheat oven to 350 degrees.
Liberally grease a 13 by 9-inch baking pan
Add the all-purpose flour, baking soda, baking powder, table salt, cinnamon, nutmeg, and allspice to a large bowl.
Whisk to combine and set aside until needed.
Add the granulated sugar, brown sugar, and eggs to the bowl of. stand mixer, fitted with the paddle attachment. Beat on medium speed until fluffy and pale, about 2 minutes.
Add the applesauce, vegetable oil and vanilla extract to sugar mixture and beat until well combined.
Pour applesauce mixture into the bowl with the dry ingredients and gently fold until just combined.
Pour the batter into the prepared baking pan. Smooth the batter with a rubber spatula.
Place the pan on the center rack of the preheated oven and bake for 35-40 minutes or until toothpick inserted into center comes out free of batter or with a few moist crumbs.
Place the pan on a wire rack to cool.