Preheat oven to 325ºF and line an 8x8 baking pan with parchment paper.

Whisk together melted butter, sugar, cocoa, vanilla, and coloring until smooth, then whisk in the eggs one at a time. Stir in the flour just until combined and then fold in the chocolate chips. Spoon into prepared pan and bake for 30 minutes.

When the brownies are cooked lift them out of the pan by gripping the parchment paper and allow them to cool for at least an hour before frosting.

For the frosting, beat together the melted white chocolate, softened butter, and vanilla until smooth. Beat in the powdered sugar a little at a time, alternating with the milk until you have reached a smooth, spreadable consistency. 

Spread the frosting on the brownies and serve or store refrigerated in an airtight container for up to a week.