Roughly chop the Oreos, set aside. Preheat oven to 350 degrees. Line baking sheets with silicone liners or spray with cooking spray.
In a large bowl, beat the butter, sugars, and dry pudding mix until smooth and creamy. Beat in the eggs and vanilla.
Whisk together the flour, baking soda, and salt in a small bowl. Add to the creamed mixture and stir gently.
Add the chocolate chips and just 1 1/4 cups of the chopped Oreos. Set aside the rest of the Oreo pieces.
Stir dough until just mixed. Roll dough into tablespoon sized balls and place 2" apart on prepared pans.
Press the cookie dough balls down slightly, then press the remaining Oreo bits into the tops of the dough. You can also press additional white chips on top.
Bake at 350° for about 10 minutes or until middles are set and edges are barely light brown.
Cool on pans for about 3-4 minutes, then carefully remove to wire cooling racks.