Chop chocolate into small pieces and melt in microwave on low heat. Once melted, add chocolate drink mix and sugar, stirring constantly until thoroughly blended. Slowly add 1 cup milk, stirring until smooth. Cool to room temperature.
Place marshmallow crème into a medium sized bowl. With an electric mixer, slowly mix in whipping cream, scraping down sides as needed. When well blended, place mixer on high and whip until soft peaks form. Place in refrigerator.
Prepare drink glasses, lightly dip the top outer rim of the glass with corn syrup. Dip or press graham cracker crumbs on the edge.
(If you have a smaller blender you may need to divide ingredients in half to blend)In blender, place the remaining 2 cup milk, room-temperature chocolate mixture and ice. Blend on high speed until smooth and the consistency of a frozen daiquiri.
Pour into glasses and top with the marshmallow creme and chocolate shavings, if desired.