In a large bowl, beat softened butter, granulated sugar, and light brown sugar on medium speed for 2–3 minutes until light and fluffy.
Mix in the egg and vanilla extract until smooth. Scrape down the bowl to blend evenly.
In a separate bowl, whisk together flour, cocoa powder, baking soda, and salt until well combined.
Slowly mix the dry ingredients into the wet mixture on low speed until just combined. The dough should be thick and slightly sticky—don’t overmix.
Stir in chopped chocolate sandwich cookies and white chocolate chips with a spatula until evenly mixed.
Cover and refrigerate for at least 30 minutes to help prevent spreading and deepen the flavor.
Preheat oven to 350°F (175°C). Line a baking sheet with parchment. Roll tablespoon-sized scoops of dough into balls and space 2 inches apart on the sheet.
Press a few extra chopped cookie pieces on top of each dough ball for a decorative crunch.
Bake for 9–11 minutes until the edges are set but the centers look slightly soft. Don’t overbake.