Preheat the oven to 350 F and line a cookie sheet with parchment paper.
Cream the butter and brown sugar together until light and fluffy, about 2 minutes using a stand mixer fitted with the paddle attachment on medium speed.
Add the egg and vanilla. Mix until just combined, then scrape down sides of the bowl.
Add the cocoa powder, baking soda, baking powder, and salt. Beat at medium-low speed until combined.
Add the flour, mix on low speed until just combined, then scrape down sides of the bowl.
Gently fold in the white chocolate chips and chopped Oreos until evenly distributed.
Using a standard ice cream or cookie scoop (3 tablespoon/1.5 ounce capacity), drop mounds of the cookie dough 2 inches apart on the cookie sheet. (At this point, you can freeze the dough balls in a single layer, then store them in a freezer-safe Ziploc for several months.)
Bake for 8-9 minutes, until the edges are just set. It is better to under rather than over bake these.
Transfer the cookie sheet to a wire rack, letting the cookies cool on the sheet for 3-4 minutes. Then transfer the cookies to the wire rack and let them cool completely.
Store at room temperature in an airtight container for up to 3 days.