Chop the chocolate into small pieces. Place it in the top of a double boiler over simmering water. Stir occasionally until melted. Add the hot chocolate mix and sugar. Stir until completely melted. Remove from heat and slowly add 1/2 cup of milk until smooth. Cool to room temperature.
In a blender, place the remaining 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 big glasses and top with whipped cream and chocolate shavings.