Preheat oven to 350 degrees F. Spray a 13x9 baking dish with baking spray (or use homemade cake release).
In a large mixing bowl, whisk together applesauce, oil, and vanilla extract. Add flour, sugars, baking powder, baking soda, salt, and spices. Mix with wooden spoon JUST until combined (batter may be lumpy).
Pour batter into prepared baking dish. Bake for 33-35 minutes, until edges begin to lightly brown and toothpick inserted into the center comes out clean.
Remove from oven and cool completely.
For the frosting, beat cream cheese with butter for 3 minutes, scraping down sides of bowl as needed.
Add in vanilla and powdered sugar, and beat an additional 3-4 minutes, until smooth and creamy.
Spread frosting over cooled cake. Cover with foil (or plastic wrap) and refrigerate until ready to serve.
Notes
Storage: keep this cake in refrigerator for best flavor (serve chilled).
If using sweetened applesauce, you can reduce granulated sugar to 1/4 cup (or keep it as is and it will be slightly sweeter than normal). Try our homemade applesauce in today's recipe!