Combine butter, cocoa powder and sugar in a small bowl set over a small pot of simmering water. Whisk until a paste forms. Add chocolate and continue whisking until chocolate melted completely. Gradually whisk in ½ cup of milk. Remove from heat and set aside to cool.

Once chocolate mixture has cooled, combine it with ice and remaining 1 cup of milk in a blender. Blend until ice is crushed and mixture is thoroughly blended. Do not over blend, as the texture should be somewhat icy.

Top with whipped cream and grated chocolate, if desired. Serve immediately.