In a blender, mix the milk and mint on high speed for 1 minute. Transfer to a saucepan and warm over medium heat, stirring occasionally, for 5 minutes. Add the white chocolate chips and stir until melted. Refrigerate until chilled.
In a blender (in batches if necessary) combine the white-chocolate mixture with the ice cream and rum. Pour into tall glasses and top with whipped cream, sprinkles and a candy cane.
The Do's and Don'ts of Cooking with Chocolate