Preheat your oven to 350°F.
In a large mixing bowl, beat the butter, granulated sugar, and brown sugar together until smooth and creamy.
Add the egg and beat until fully combined.
Reduce the mixer speed and mix in the flour, crushed Oreos, baking soda, and baking powder just until no dry streaks remain.
Divide the dough into 12 portions. Roll into balls and gently flatten into thick discs.
Chill the shaped dough balls in the refrigerator for 30 minutes. This step helps the cookies hold their shape and bake up soft and chewy.
Bake for 11 minutes, until the edges are set and the centers still look slightly soft.
Let the cookies cool on the pan for several minutes before transferring to a wire rack. Cool completely before frosting.
For the frosting, beat the butter, powdered sugar, and finely crushed Oreos together until smooth and fluffy. Spread or pipe onto cooled cookies.