Preheat the oven to 350ºF (177ºC) and prepare three 6-inch or two 8-inch cake pans by spraying the sides with cooking spray and fitting the bottoms with a wax paper or parchment cake circle.
Roughly chop the Oreos into pieces that are about 1/4 inch. There can be some bigger pieces in there, but keep in mind that the bigger pieces may sink slightly in the baking process. Once the Oreos are chopped, set them aside.
Add the sifted cake flour, baking powder, baking soda, and salt into a bowl and whisk to combine. Set aside.
In the bowl of your stand mixer fitted with the paddle attachment (or with a hand mixer and large bowl), cream the butter on high for two minutes. Add the sugar and vegetable oil and continue to mix on high for another two minutes, scraping down the bowl and paddle as needed. Add the eggs one at a time, mixing thoroughly after each addition. Add the vanilla and sour cream and mix for one minute on high speed, scraping down the bowl and paddle once more.
Turn the mixer off and add the dry ingredients all at once. Mix on low speed until just combined, then add the milk slowly and mix until just incorporated. Fold in the chopped Oreo cookies.
Pour the batter into the prepared cake pans and bake for 30-35 minutes, until a toothpick inserted comes out clean. Cool the cakes in the pan for five minutes before removing and continuing to cool on a wire rack or flat surface.