Preheat the broiler. On a foil-lined baking sheet, place half of the marshmallows in a single layer; broil until charred, 3 minutes. Transfer to a plate to cool. Place the remaining marshmallows on the sheet and roast just until toasted, 2 minutes; let cool.
In a blender, mix the milk, ice cream, sour cream and charred marshmallows. Serve topped with the toasted marshmallows.