Set four tall glasses in the freezer to chill.
Melt chocolate in a medium saucepan over low heat, stirring constantly, just until completely melted. Remove from heat. Stir in cocoa and granulated sugar to blend.
Slowly stream in 1/2 cup milk, whisking constantly. If chocolate begins to seize up, return saucepan to low heat, whisking again until melted.
Off of heat, stir in remaining 1 cup milk and the rum. If needed, chill mixture in freezer about 15 minutes until cold (this prevents separation when blending with ice).
In a bowl with a mixer, beat cream and powdered sugar to medium peaks.
Put ice and chocolate mixture in a blender and whirl until smooth but still icy. Pour into chilled glasses and top with whipped cream and shaved chocolate.
* To shave, set chocolate on a piece of waxed paper and microwave until slightly warm, 10 to 15 seconds, then shave with a vegetable peeler.