Make cookies: Preheat oven to 350°F. Line a baking sheet with parchment paper.
In the bowl of a stand mixer fitted with a paddle attachment, cream the butter and sugars on medium speed for 1-2 minutes. Add the egg and vanilla extract and mix to combine.
Add the crushed Oreo crumbs, all purpose flour, cornstarch, baking soda, and salt. Mix until the dry ingredients are partially combined. Then add the white chocolate chips and roughly chopped Golden Oreo pieces. Mix to combine.
Using a ¼ cup sized cookie scoop, portion the dough into 9 balls. Press down on the balls with the palm of your hand until the cookies are roughly 1” thick. Bake in the preheated oven for 8-9 minutes. Remove and let cool on the pan for 10 minutes before transferring them to a wire rack to cool completely.
While cookies are cooling make the frosting. In a bowl of a stand mixer fitted with a whisk attachment, mix the cold cream cheese on medium speed for 1 minute. Add the powdered sugar and vanilla extract and mix on medium speed for 2 minutes, scraping down the sides of the bowl to make sure everything is combined.
With the mixer on, slowly pour the heavy cream down the side of the bowl, into the cream cheese mixture. Pouring it along the edge of the bowl will prevent the cream from hitting the whisk and splashing everywhere. Turn the mixer to medium high speed. The frosting should thicken and take shape. Mix for about 30 seconds, but keep a close eye on the frosting as over-mixing it can cause the mixture to get soupy.
You'll know the frosting is finished when you stop the mixer, dip the whisk attachment in the frosting and turn it upside down. The frosting should keep its shape and hold its' peak. If yours hasn't reached this stage, mix in 15 second intervals and test again.
Place the frosting in a piping bag and pipe the frosting atop each cookie. Alternatively, you can use a knife and smear the frosting overtop each cookie. If you don’t eat the cookies the day they are made, you should store leftovers in the fridge (due to the whipped cream cheese frosting).