To prepare the cookies, preheat the oven to 350 degrees and prepare two baking sheets by covering them with parchment paper or silicone baking mats.
In the bowl of a stand mixer or a large mixing bowl, cream together the butter, brown sugar, sugar, and vanilla on medium speed until smooth and fluffy, about 2 minutes. Scrape the sides of the bowl and add the eggs one at a time, beating on low until combined. In a separate bowl, combine the flour, salt, and baking soda and then add this mixture to the butter and sugar. Stir on low just until combined. Scrape the sides of the bowl and then fold in or stir on low the white chocolate chips and Oreo pieces .
Scoop out 3 tablespoon sized balls of dough (I used a large cookie scoop) for large cookies or 1-1/2 tablespoon sized balls of dough (medium cookie scoop) for smaller cookies. Roll the balls quickly in your hands for smooth cookies. Space out dough balls three inches apart (2 inches if using smaller dough balls) on the baking sheet and bake in the oven for about 10-12 minutes (9-10 minutes for smaller dough balls), or until the edges are set and the centers are beginning to set. The cookies may look slightly underdone when you pull them out of the oven. Allow to cool briefly before enjoying!
To prepare the optional filling, cream together the butter and shortening in the bowl of a stand mixer. Add the confectioner’s sugar and vanilla and mix on low until incorporated. Increase the speed to medium and beat until smooth and creamy. You can add a bit of milk or water to thin out if desired. Spread the filling in between two cookies and sandwich together!