Mix milk and sugar together until sugar is dissolved. Add cream, vanilla extract, and peppermint oil, and mix well.

Pour into your ice cream maker and follow the manufacturer's directions.

While ice cream is churning, crush candy canes. (I put mine in a blender and pulse until finely chopped).

Put white chocolate chips and vegetable oil in a small saucepan on the stove. Heat over medium-low until fully melted and smooth. Remove from heat.

About five minutes before ice cream is done, add the crushed candy canes. Then transfer to an airtight container, drizzling in the melted white chocolate in layers. (Put some ice cream in the container, then drizzle chocolate, then add more ice cream and drizzle more chocolate, etc.) Chill for several hours.