Whip the cream with the whisk attachment of a stand mixer (or a handheld electric mixer) until soft peaks form; beat in the powdered sugar and vanilla.
Put the milk, cocoa powder, and sugar in a medium saucepan over medium heat. Cook, whisking constantly, until the cocoa powder is fully dissolved and the mixture begins to steam, about 5 minutes.
Divide the liqueur between two mugs. Add the hot chocolate to the mugs, and top with the whipped cream. Serve hot.
