Preheat your oven to 350°F (175°C). Spray a 9x13-inch baking dish with nonstick cooking spray.

In a large bowl, whisk together the flour, baking soda, baking powder, salt, cinnamon, ginger, nutmeg, and allspice. Make a well in the center and set aside.

In another large bowl, combine the granulated sugar, brown sugar, and eggs. Beat with an electric mixer on medium speed until pale and fluffy, about 2 minutes.

Add the applesauce, vegetable oil, and vanilla extract to the sugar mixture. Mix until well combined.

Pour the wet mixture into the well in the dry ingredients. Gently fold until no flour streaks remain. Do not overmix.

Pour the batter into the prepared baking dish and spread evenly.

Bake for 30 to 40 minutes, or until a toothpick inserted in the center comes out clean. Let the cake cool completely on a wire rack.

For the frosting, beat the cream cheese and softened butter together until smooth. Add the powdered sugar and vanilla extract, then whip on high speed until light and fluffy.

Spread the cream cheese frosting evenly over the cooled cake. Slice and serve.