Preheat oven to 350 degrees F. Butter half sheet pan (18x13-inch) and line with parchment paper. Set aside.
In the bowl of a stand mixer fitted with paddle attachment, cream together butter, sugar, and vanilla until smooth and fluffy.
In a separate bowl, whisk together cake flour, baking powder, and salt. With the mixer running on low, add flour mixture in three additions alternating with milk. Start and finish with flour mixture. Scrape down bowl as needed to ensure thorough mixing. Fold in chopped Oreo cookies.
In a large clean bowl, whip egg whites using an electric hand mixer. Start at low speed until egg whites are foamy. Increase speed and continue to whip to stiff peaks. Gently fold egg whites into cake batter in three additions. Cake batter will loosen as you add the egg whites.
Transfer cake batter to prepared sheet pan. Spread into an even layer. Bake for 25-30 minutes, rotating pan midway through baking, until toothpick inserted in center of cake comes out clean.
Allow cake to cool to room temperature in pan. Use a mini offset spatula or butter knife to loosen cake from pan. Use a 9-inch parchment round as a guide to cut out one layer of cake. Fold parchment round in half and cut out two half rounds using remaining cake. Immediately use cakes for assembly, or wrap in plastic wrap for later use.Note:There will be plenty of leftover cake scrapes. Save this to garnish the cake, or use for other recipes- such as cake balls, milkshakes, cookies.