Pour in vegetable oil and applesauce, set mixer with paddle attachment and blend until well combined (occasionally stop and scrape bowl throughout mixing process).
Divide mixture evenly among two prepared baking pans. Bake in preheated oven until toothpick inserted into center comes out clean, about 28 - 32 minutes.
For the frosting: In the bowl of an electric stand mixer fitted with the paddle attachment cream butter and cream cheese until smooth.
Mix in powdered sugar and vanilla and mix until well blended***.