Preheat oven to 325. Then prepare a 9x9 inch baking dish by spraying lightly with non-stick cooking spray. Cut a piece of parchment paper to the width of the bottom of your pan but long enough to extend over the edge of the right and left sides of the pan.
Sift brownie mix to eliminate any clumps. Then add melted butter, water eggs and 1/4 teaspoon of peppermint extract. Mix by hand just until dry ingredients are incorporated.
Pour into prepared pan. Bake according to package directions. Let cool.
Beat together cup powdered sugar and 1/2 cup softened butter. Mix in another 1/2 cup powdered sugar, then add 1 tablespoon milk and peppermint and vanilla extracts. Spread frosting evenly over the top. Place brownies in fridge to firm up frosting.
Bring 1/3 cup milk and 1/4 cup butter almost to a boil over medium heat. Pour chocolate chips into a heat-safe bowl. Pour milk mixture over the top and let sit for 5 minutes. Add 1 teaspoon vanilla and whisk until smooth. Use an electric mixer to beat in 1 cup powdered sugar until frosting is smooth. This will make more than you need, so remove 1/4-1/3 cup frosting and reserve for another use.
Spread chocolate frosting over the white frosting on the brownies.
Place back in fridge for about an hour to firm up frosting. Remove brownies from pan and slice into 9 pieces.