Chop the chocolate into small pieces and place it in the top of a double boiler over simmering water, stirring occasionally until melted. Add the cocoa and sugar, stirring constantly until thoroughly blended. Remove from heat and slowly add 1/2 cup of the milk and stir until smooth. Cool to room temperature.

In a blender place the remaining 1 cup of milk, the room temperature chocolate mixture, and the ice. Blend on high speed until smooth and the consistency of a frozen daiquiri. Pour into a giant goblet and top with whipped cream and chocolate shavings.