Preheat your oven to 350 degrees F. and prepare 3 6-inch or 2 8-inch cake pans by rubbing generously with vegetable shortening and dust with flour.
In the bowl of a stand mixer equipped with a whisk attachment, add the flour, sugar, baking powder, and salt. Whisk until combined.
With the mixer on low speed, add the butter in 1 tablespoon at a time. Mix until you have a dry sand- like consistency.
Add in the vegetable oil, sour cream, vanilla, and milk and whisk until combined and smooth.
Add the egg whites in last and mix until the whites are just mixed into the batter.
Fold in the Oreo pieces last.
Divide the batter evenly between the cake pans. If you want to be precise, measure the batter and divide by 3. (Or 2 if you are using 8-inch pans)
Bake for 35-40 minutes checking around the 30-minute mark for doneness. The cakes are done when you can insert a toothpick into the thickest part of the cake and the toothpick comes out clean.
Remove from the oven and allow the cakes to cool about 10 minutes in the pan, then invert onto a wire rack to cool completely,