Preheat the oven to 325. Add the butter, shortening, and brown sugar to a bowl. Using a stand mixer or hand mixer, beat it all together until it is fluffy, about 2 minutes. Add the egg and vanilla extract and mix again.
Add the flour, cocoa powder, salt, and baking soda and mix until just combined. Before adding the oreos, make sure they are very crushed -- really powdery and fine. Add the oreos and mix until they are evenly distributed in the dough. The dough will be a dark brown color.
Using a large cookie scoop, scoop up to 6 dough balls onto a normal sized baking sheet. Bake at 325 for 8-10 minutes. When the cookies are out of the oven, let them cool completely before frosting them.
While the cookies are cooling, make the frosting. Add the butter to a large bowl. Using a hand mixer or stand mixer, beat the butter until it is smooth and lighter, about 3 minutes. Add the powdered sugar and beat another 3 minutes.
Add the pumpkin puree, pumpkin pie spice, and vanilla and mix an additional 3 minutes. The frosting should be smooth. Frost each cookie and crumble some oreo on top. Enjoy!