In a large bowl, or in the bowl of a stand mixer with the paddle attachment, beat the cream cheese on med-high until creamy.
Scrape down the bowl and then add the butter. Beat well - until pale and smooth.
Mix in the sugar and brown sugar, beating on high for about 2 minutes. This ensures the butter blends into the creamy mixture and the texture isn't gritty.
Add the egg and vanilla - mixing until just combined, taking care to not overmix. Scrape the sides of the bowl and then mix for a few seconds longer. (no more than 1 minute total)
Slowly add the flour, cornstarch, baking soda, and salt, mixing in a little at a time on low, just until combined.
Use a large rubber spatula to fold in the crushed Oreos, chocolate chunks, and white chocolate chips.
Cover the bowl with plastic wrap and refrigerate for at least 2 hours. I usually leave mine overnight and bake the next morning. Let the dough sit at room temperature while you preheat the oven if chilling overnight.
Preheat the oven to 350 degrees.
Line baking sheets with parchment - set aside.
Roll the dough into balls - using about 1.5 tbsp of dough for each. Make the dough slightly oblong, so they are taller than they are wide - so they bake nice and thick. Place on the prepared baking sheets about 2" apart.
Bake 12-13 minutes, or until the edges are golden brown but the centers still look a bit soft.
Remove and let cool for 5 minutes on the baking sheet before transferring to a wire rack to cool completely. You can press some extra chocolate chips, chocolate chunks, or extra cookie pieces into the tops of the cookies before they cool if desired.
Keep in an airtight container at room temperature for up to 5 days.