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Tip the butter, caster sugar and flour into a food processor with a pinch of salt. Blitz until the mixture looks sandy. Add the egg yolk, vanilla extract and 1 tbsp cold water, and blitz again until the dough clumps together. Tip onto a surface and knead briefly until all the flour is incorporated. Shape into a disc, wrap and chill for 30 mins, or up to two days. Will keep frozen for two months. If chilled for longer than 1 hr, leave at room temperature for 10 mins before rolling out.
Line two baking trays with baking parchment. Heat the oven to 180C/160C fan/gas 4. Roll the dough out on a lightly floured surface until it’s a little thicker than a £1 coin. Stamp out stars using the cutter, then carefully lift onto the baking trays and gently push a lolly stick into the base of each. Bake for 12-15 mins until golden at the edges (you may need to do this in batches). Cool on the trays for 5 mins, then transfer to a wire rack to cool fully.
Combine the icing sugar with 2-3 tsp water to make a thick icing. Spread this over the biscuits, then scatter with sprinkles. Leave to set for 1 hr, then tie ribbons around the wands, if you like. Best eaten within 24 hrs, but will keep in an airtight container for three days.