Preheat the oven to 325°F. Line a baking sheet with parchment paper and set it aside.
Using a handheld mixer, combine the butter, shortening, sugar, eggs, and vanilla in a large bowl.
To the butter mixture, add the flour, cornstarch, salt, and baking powder. Mix until the ingredients are incorporated.
Add the crushed Oreo cookies to the dough. Mix on high speed to finely crush the Oreo pieces.
Using a ¼ cup (or 2 scoops of dough with a 2 tablespoon scoop), add the dough out onto a lined cookie sheet, about 2 inches apart. (I usually put 6 cookie dough balls onto each cookie sheet.) Use your hands to shape and slightly flatten each ball of dough. (If you used 2, 2 tablespoon scoops, combine the scoops of dough into one large cookie.)
Bake for 12 minutes for a softer cookie, up to 14 minutes for a firmer cookie.
Remove cookies from the oven and let them cool completely before frosting with the buttercream.
Store frosted cookies in the refrigerator. Serve cookies cold.