Preheat oven to 350 degrees. Line 12 muffin cups with paper liners.
In the bowl of a stand mixer, beat the butter and sugar until light and fluffy. Add the eggs, one at a time, beating well after each addition. Beat in the vanilla extract.
In a medium size mixing bowl, whisk together the flour, baking powder, and salt.
With the mixer on low speed, combine ingredients. Alternate the flour mixture and milk, in three additions. Do not overmix. Remove bowl from stand and scrape down the sides of the bowl as needed, making sure to incorporate any dry ingredients. Fold in crushed Oreo pieces.
Place one whole Oreo on the bottom of each liner. Evenly fill cups with the batter (about ⅔ of the way full). Place in oven and bake for about 16-18 minutes, or just until a toothpick inserted into a cupcake comes out clean (do not overbake, results will be dry). Remove from oven and allow to cool in pan for 5 minutes. Remove cupcakes and transfer to a wire rack to cool completely (30 minutes or so).