For the Brownies: Preheat oven to 350 degrees and grease a 9×13 baking pan. In a large glass bowl combine sugars and butter and microwave for a minute (until butter is melted). Stir in the vanilla and eggs until complete combined. Add cocoa powder, flour, salt, and baking soda and mix until smooth. Fold in chocolate chunks and peppermint chips. Bake for 30 minutes and cool completely before frosting.
For the Peppermint Bark Topping: Cream butter and cream cheese together until very fluffy (about 5 minutes in a stand mixer). Add powdered sugar and vanilla and mix to combine. If you don’t sift the sugar (and I do not) it will be lumpy. That’s totally okay. Add melted white chocolate and mix in completely. Add milk. Smooth frosting on top of cooled brownies and sprinkle with roughly chopped peppermint bark, or with peppermint chips.
Refrigerate frosted brownies for an hour to make cutting easier (though I won’t blame you if you dig right in).