In a small bowl, whip cream and Gran Classico into soft peaks and chill until using (up to 2 hours). Or, combine cream and liqueur in a whipped-cream siphon*.
Pour crème de cacao, milk, and both crèmes de menthe into a strainer-topped cocktail shaker or glass. Fill with ice. Shake 30 seconds, then pour into a martini glass through a second, small strainer to catch any ice chips.
Top with enough Gran Classico whipped cream, trying not to disturb liqueur underneath, to come just below rim of glass. Lay a small piece of paper over 2/3 of glass, then grate chocolate over exposed 1/3 of cocktail. Remove paper and serve.
*The sturdy iSi Gourmet Whip whips both cold and hot mixtures ($150; a pack of 10 iSi cream charger cartridges, $15; williams-sonoma.com). It comes in handy if you're topping multiple cocktails.