Preheat the oven to 350F (180C). Line the bottom of two 9-inch cake pans or two 8-inch cake pans with 2 inch sides with parchment paper and lightly grease the sides. If using a 9x13 inch pan, simply grease the pan.
In a medium bowl, sift together the flour, baking powder and salt. Then whisk together. Set aside.
In a large bowl, beat the butter until creamy. Add in the sugar a little at a time and continue beating until all the sugar is incorporated. Then beat in the oil and vanilla extract.
Beat in the egg whites one at a time, stopping as soon as you no longer see pieces of egg. Turn off the mixer and scrape down the bottom and sides of the bowl a few times about ½ way through.
With the mixer on a low speed, beat in about ⅓ of the dry ingredients. Then beat in about ½ of the milk. Stop mixing as soon as each addition is incorporated. Turn off the mixer and scrape down the sides and bottom of the bowl.
Beat in about ⅓ more of the dry ingredients, followed by about ⅓ more of the milk. Then beat in the rest of the dry ingredients until just combined, or whisk in by hand to avoid overmixing.
Fold in the chopped Oreo cookies using a rubber spatula or wooden spoon.
Distribute the batter evenly between your prepared cake pans.
Bake cakes in the middle of the preheated oven - be sure that both cake pans are on the same baking rack. Two 9-inch round pans will take about 35-40 minutes to bake. Two 8-inch round cake pans (with 2-inch sides) will take about 40-45 minutes. When the cakes are done baking, an inserted toothpick should come out clean and the cakes should be starting to pull away from the edges of the pan.
Cool the cakes in the pans for at least 15 minutes, then carefully transfer to a wire rack to continue cooling.