In the bowl of a stand mixer, beat butter and sugar until light and fluffy. Add vanilla and eggs and beat until smooth.
Add flour, baking soda, salt and cocoa and beat until combined. Add chocolate chips and beat on low until combined. If cookie dough is still a bit sticky, refrigerate for 1 hour.
Prepare 24 muffin cups: use a regular muffin pan, well-greased or use silicone muffin cups ungreased (a dark pan will result in crispier edges, a silicone pan will result in softer edges)
Drop a rounded teaspoon of cookie dough in each muffin cup. Press down with your fingers and work dough up the edges just slightly. Place one Oreo in the middle, pressing down gently. Scoop another rounded teaspoon of dough and flatten it into a circle the size of the muffin cup. Place on top of each Oreo and press down the sides gently to surround the Oreo.
Bake for 12-14 minutes until centers appear set. Set aside to cool.
In a small bowl, melt white chocolate and milk in the microwave in 30 second intervals, stirring in between. Set aside to cool slightly (15-20 minutes)
With an electric mixer, beat butter until smooth. Add cooled chocolate mixture and vanilla and beat until combined.
Add powdered sugar, one cup at a time, beating until combined after each addition. Beat on high 2-3 minutes until light and fluffy.