Dessert, cocktail, or both? However you look at this grown-up drink, the answer is yes. The rich, sweet caramel notes found in rum nicely complement the chocolate, but bourbon, whiskey, or stout make good alternatives.
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.