Make the Cookie Cake: Preheat oven to 350°F. Grease a 10-inch springform pan with cooking spray and set aside.
Cream together butter and sugars for 3 minutes. Add egg and egg yolk, and beat for an additional 2 minutes. Add vanilla extract and milk, and beat for another 2 minutes.
In a separate bowl, mix together flour, milk powder, salt, baking soda, and baking powder. Add to butter-sugar mixture, and mix on low until fully combined, about 1 minute. Add chocolate chips and fold into dough until evenly distributed.
Spread dough evenly into prepared springform pan, and bake until dough is just cooked through and the surface is lightly browned, 23 to 25 minutes. Allow to cool completely before removing from pan and frosting, about 1 hour.
Make the Chocolate Frosting: Beat all ingredients together until smooth and lightened in color, about 3 minutes.