Preheat the oven to 160°C fan bake.
Cut the butter into cubes and add to the bowl of a mixer. Add the icing sugar and vanilla, then beat for a few minutes until pale and fluffy.
Sift in the cornflour and plain flour. Mix on low until it forms a fine crumb. Press the dough into a ball and knead a few times on a floured bench.
Roll the dough out to a 1cm thickness. Use a round or square cookie cutter to cut into shapes.
Carefully transfer the cookies to a lined baking tray and bake for 12 minutes until they start to brown on the edges but are still soft.
Remove from the oven and leave to cool on the trays.
To make the icing: Add the butter and milk to a large glass bowl, microwave until the butter has melted, then whisk together. Sift in the icing sugar and whisk until smooth. Add extra milk if needed to make a smooth icing.
Spoon a teaspoon of icing onto each cooled cookie, spreading it to the edges. Dip each one in the sprinkles until they are all coated.
Once the icing is set, serve or store in an airtight container for up to 5 days.
