Melt the butter in a pan, leave to cool for a few minutes, then pour into a large bowl. Whisk in both sugars and the egg until combined.
Use a wooden spoon to mix in the flour and baking powder until the mixture comes together into a dough. Beat in the peppermint extract, then fold in the chocolate chips. Wrap well and chill for 1 hr.
Heat the oven to 180C/160C fan/gas 4. Divide the cookie dough into 16 pieces (about 45g each), and roll each piece into a ball using your hands. Arrange the dough balls on two baking trays lined with baking parchment, well spaced apart, and press them down slightly. Bake for 10 mins, then remove the trays from the oven and tap sharply on a work surface to flatten the cookies. Return to the oven and repeat this after 2 mins, then again after another 2 mins (they should bake for 14 mins in total). Leave to cool completely on the trays.
Melt the chopped chocolate in a heatproof bowl over a pan of simmering water, or in 15-second bursts in the microwave, stirring between each burst. Line a baking tray with baking parchment. Dip one half of each cookie into the melted chocolate, place on the lined tray and sprinkle with candy cane pieces. Transfer the tray to the fridge and chill for 30 mins to set.