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Put everything for the shortbread in a food processor with a pinch of salt and pulse until the mixture comes together to form a dough. If it won’t come together, take it out of the processor and bring it together with your hands. Tip onto a lightly floured work surface and shape into a log about 5cm in diameter. Wrap in cling film and chill for 1 hr.
Heat oven to 140C/120C fan/gas 2. Line two large baking sheets with baking parchment. Cut the log into about 32 rounds 6mm thick and line them up on the sheets. Bake for 22-25 mins. Leave to cool a little, then carefully remove and put on a wire rack to cool completely.
To make the buttercream, blitz all the ingredients in a food mixer or beat with an electric whisk, adding enough of the milk to make a soft mixture. Sandwich the shortbread together with the buttercream.