In a small saucepan over medium heat, bring milk to a simmer. Whisk in sugar and cocoa powder and stir until no lumps remain. Stir in chocolate chips and vanilla and cook, stirring occasionally, until the chocolate is melted. Turn off heat.
Pour hot fudge in a small dish. Dip the tops of mugs in hot fudge to rim the cup, then pour hot chocolate into each mug. Top with a scoop of CORTINA ice cream and garnish with chocolate shavings.