Preheat oven to 350 degrees. Use parchment paper or a silpat mat to line a baking sheet.
Cream the butter in a stand mixer until smooth and creamy. Add brown sugar and sugar and mix until light and fluffy (about 2-3 minutes). Scrape the sides of the bowl to make sure all ingredients are blended.
Add in the egg and vanilla and beat until combined.
Add in flour, Oreo crumbs, baking soda, and salt. Mix until all combined.
Scoop out about 1 TBSP of dough and roll into balls in your hand. Place dough balls on prepared pan. (12 cookies to a pan) Press down dough slightly to form thick discs.
Bake for 8-10 minutes. Allow cookies to cool on the pan for 5-10 minutes before transferring to a cooling rack. Cool cookies completely before making sandwiches.
HOW TO MAKE COOKIE DOUGH FROSTING
Beat the butter in a stand mixer (or with a handheld mixer) for about 1 minute until creamy.
Add in the brown sugar and mix until combined.
Add in vanilla, flour and salt and mix on low speed. Add milk and continuing mixing until everything is incorporated. Then switch your mixer to high speed and beat for 2 minutes until the frosting is creamy and smooth. Stir in chocolate chips.
Sandwich about 1-2 tbsp of cookie dough frosting (more or less depending on your taste!) between two cooled cookies.