Place butter and chocolate in the top of a double boiler over simmering water. Stir frequently, scraping down the sides with a rubber spatula to avoid scorching, until melted and smooth, about 5 minutes. Let cool for a few minutes.

Combine confectioners' sugar, 1/2 cup milk, and vanilla in a large bowl. Use an electric mixer and slowly blend in melted chocolate mixture. Add remaining 1/4 cup milk, a little at a time, until desired consistency is achieved.

Let stand until frosting thickens enough to spread, 5 to 10 minutes.