Break the bittersweet chocolate into pieces and place it in a medium bowl set over a pot of simmering water, stirring occasionally, until just melted. Off the heat, whisk in the cocoa powder, sugar, and half-and-half and set aside.
Place 3/4 cup of the milk, 2 tablespoons of the coffee liqueur, 1/2 teaspoon of the vanilla, and half of the chocolate mixture in a blender. Add 2 cups of the ice and blend for several minutes, until the mixture is thick but not icy, like a frozen daiquiri. (You can add a little more milk or ice if it's too thick or too thin.) Pour into two (10-ounce) ice cream soda glasses and top each with a generous dollop of whipped cream and some grated chocolate. Repeat for the second two glasses. Serve ice cold with straws and long-handled spoons.
Place the cream, sugar, and vanilla in the bowl of an electric mixer fitted with the whisk attachment and beat on high speed until the cream forms very soft peaks (don't overbeat or the cream will be too firm).
Ina halved the recipe for taping.
This recipe has been updated and may differ from what was originally published or broadcast.
"Cooking for Jeffrey" by Ina Garten © Clarkson Potter 2016. Provided courtesy of Ina Garten. All rights reserved.