Combine milk and cream (or half & half) in a large sauce pan over medium heat. Heat, stirring occasionally, until hot throughout. Reduce heat to medium-low.
Add white chocolate and stir until completely melted and incorporated. Stir in vanilla and peppermint extract. Serve with a candy cane on the side of the glass or crushed candy canes around the rim.
Notes
Pro tip: serve from a slow cooker set on low to keep it warm!