Put about an inch of water in the bottom of a medium pan and place it over medium-low heat and let it heat to steaming, but not boiling. Place the butter and chocolate in a heat-proof bowl— I like to use the bowl from my KitchenAid Mixer. Put that on top of the pan of steaming water and allow it to melt, stirring occasionally with a rubber spatula.

Once the chocolate and butter have completely melted, remove them from the heat. Add in the powdered sugar. Use the whip attachment on your stand mixer, or use a hand mixer to beat in the powdered sugar. Start on low to avoid a powdered sugar explosion, then slowly turn up the speed as it is incorporated. The mixture will look lumpy and thick. Add in the milk and vanilla and beat until the frosting is smooth and fairly runny. Be sure to scrape the sides of the bowl and beat in any lumps or splashes until smooth.

Allow the frosting to cool for an hour or two. It should thicken to a nice consistency for piping and frosting cake layers.