Melt together over low heat just 1 cup of the milk with all the chocolate. Bring to a slow, rolling boil.
While the mixture is boiling, use a separate saucepan to heat the rest of the milk with the white sugar. Whisk it for several minutes or until the sugar has dissolved. Combine with chocolate mixture and keep on medium heat.
On high speed, beat together the whipping cream and vanilla. Beat in the confectioners' sugar, and continue beating briefly just until the whipped cream forms soft peaks.
Remove the chocolate from the heat, and divide into cups. Top each cup with a generous dollop of whipped cream and serve. Enjoy!