Preheat oven to 350° degrees F. Line a 9" or 10" springform pan with a parchment paper round, then liberally grease the pan. Set aside.
In a large bowl of a stand mixer, cream together the butter, brown sugar and granulated sugar until fluffy and light, about 1 minute. Add in the egg, followed by the egg yolk, mixing well in between additions. Add in the vanilla, red food coloring and vinegar and mix briefly. Lastly, add in the baking soda, cornstarch, salt, cocoa powder and flour and mix on low speed until just combined. Stir in the white chocolate chips.
Spread the cookie dough into an even layer in the prepared pan. Bake for 25-30 minutes or until a toothpick inserted near the center comes out clean. Cool the cookie cake in the pan completely.
For the frosting: In a large bowl, beat together the butter, cream cheese and vanilla with a handheld electric mixer until creamy, about 30 seconds. Gradually add in the confectioners' sugar, beating on low speed until a soft and spreadable frosting consistency is achieved.
Remove the cookie cake from the springform pan. Pipe a border of cream cheese icing around the edge of the cookie cake's surface. Garnish with sprinkles, if using.