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Heat the oven to 180C/160C fan/gas 4. Beat the butter with the icing sugar using an electric whisk until the mixture is light and fluffy. Beat in the vanilla and flour (the mixture will stiffen and look like crumble as the flour is added). Add the milk and keep beating until the mixture softens and sticks together (add some more milk if needed).
Scoop the dough into a piping bag fitted with a large star nozzle and pipe swirled rings onto a baking sheet lined with baking parchment. If the mixture is too stiff to pipe easily, you can roll the dough into balls and put them on the sheet instead. The cookies will spread as they cook so don’t worry if there’s a small gap in the centre of the piped swirls, but ensure there is enough space between each cookie so they don’t stick together as they cook. Bake for 15 mins, or until lightly golden, then transfer to a wire rack to cool.
Put the white and milk chocolate in separate bowls and microwave each in 20-second bursts until melted. Dip the cookies into each chocolate, milk at one end and white on the other, then let any excess drip off before returning them to the rack to set. Will keep in an airtight tin for up to four days.