Place milk and cookies in a medium bowl. Stir well and steep for 20-25mins. Strain into a small bowl using a fine mesh sieve. Gently press the milk out of the cereal using a spatula or wooden spoon. Separate out 1 cup of the cookie milk for the cake. Save the remainder to drizzle onto the cake layers.
Preheat oven to 325F. Grease and flour three 6" cake rounds and line with parchment.
In a medium bowl, whisk flour, baking powder, and salt until well combined. Set aside.
Using a stand mixer fitted with a paddle attachment, cream butter and sugars on med-high until pale and fluffy (3mins).
Reduce speed and add eggs one at a time fully incorporating after each addition. Add vanilla.
Alternate adding flour mixture and cookie milk, beginning and ending with flour (3 additions of flour and 2 of milk). Fully incorporating after each addition.
Fold in chocolate chips and mix until just incorporated. Do not overmix.
Bake for 35-40mins or until a toothpick inserted into the center comes out mostly clean.
Place cakes on wire rack to cool for 10mins then turn out onto wire rack. Allow cakes to cool completely.
Place egg whites and sugar into the bowl of a stand mixer, whisk until combined.*
Place bowl over a double boiler on the stove and whisk constantly until the mixture is no longer grainy to the touch (approx. 3mins).
Place bowl on your stand mixer and whisk on med-high until the meringue is stiff and cooled (the bowl is no longer warm to the touch (approx. 5-10mins)).
Switch to paddle attachment. Slowly add cubed butter and mix until smooth.
Add vanilla and whip until smooth.**
Place chopped chocolate and cream into a microwave safe bowl. Stir to combine. Microwave for 20 seconds, stir. Microwave in 10 second intervals, stirring in between, until ganache is smooth and silky. Set aside to cool and thicken slightly before using on cake.
Trim the tops of the cake layers if needed or poke holes in them with a bamboo skewer. Drizzle or brush each cake layer with about 2 Tbsp cookie milk.
Place one layer of cake onto a cake stand or serving plate. Top with approximately 2/3 cup of buttercream. Repeat with remaining layers and crumb coat the outside leaving the sides of the cake exposed. Use a bench scraper to smooth out the sides and top of the cake. Chill for 20mins.
Using a teaspoon, apply ganache near edges like so to create the drips. Pour some ganache on the top of the cake and spread with an offset spatula.
* Ensure there is NO trace of egg yolks in your whites and that your mixer bowl and whisk is completely grease-free or your meringue won't stiffen. ** The buttercream may look like it's curdled at some point. Keep mixing until it is completely smooth.***RECIPE REVISED: Aug 8, 2019*** - Revised the cake recipe to add in brown sugar and also cookie milk for more of that Milk & Cookies flavor.***RECIPE UPDATED: Feb 5, 2017*** - A few of you mentioned that you found the cake dry and dense so I've updated it to a lighter, more fluffy version that hopefully everyone should find delicious!