Warm the milk in a small saucepan over medium-low heat, whisking often, for 3 to 4 minutes, until small bubbles form at the edge of the pan but it never reaches a boil.

Add the white chocolate syrup and whisk for about 1 minute, until it dissolves completely and the milk turns a pale ivory color.

Whisk in the peppermint extract.

Taste the mixture before adding more; peppermint extract goes from pleasant to toothpaste fast, so adjust 1 drop at a time.

Pour the hot mixture into a heatproof mug.

Top with whipped cream.

Sprinkle the crushed candy cane over the whipped cream and serve right away, while it is still steaming.