In a medium saucepan, heat milk over medium-low heat until just starting to bubble around the edges. 

Whisk in milk chocolate, dark chocolate, sugar and vanilla until smooth and combined. Remove from heat and cover with a lid to keep warm.

In a small saucepan, heat ½ cup heavy cream over medium heat until hot but not boiling. 

Add white chocolate and whisk until smooth. Let cool to room temperature. 

Beat remaining cup of heavy cream to soft peaks with an electric mixer. Add white chocolate mixture and beat until stiff peaks form.

Divide hot chocolate between 6 mugs and top with white chocolate whipped cream.