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Warm the butter and milk together in a pan over a low heat until the butter has melted. Remove from the heat and leave to cool for a few minutes until just tepid. Meanwhile, tip the flour, sugar, yeast, cardamom and ½ tsp salt into the bowl of a stand mixer fitted with a dough hook. Pour the warm milk and butter mixture into the dry ingredients and stir to combine, then knead for 5 mins until you have a smooth dough. Or, knead by hand in a large bowl for 8-10 mins. Cover and leave to prove in a warm place until the dough has doubled in size, about 1 hr 30 mins.
For the custard, warm the milk, cream and vanilla in a pan over a low heat until beginning to steam. Whisk the egg yolks, sugar and cornflour together in a large heatproof bowl, then gradually pour in the hot milk mixture, whisking continuously. Pour the mixture back into the pan and heat gently, stirring occasionally for 5-6 mins until thickened into a custard. Pour into a heatproof bowl, cover with a circle of baking parchment and leave to cool, then chill until ready to use.
Line one or two baking trays with baking parchment. Knock the dough back, then divide into 12 pieces (about 70g each) and form into smooth balls. Arrange over the trays, spacing well apart, then cover and leave to prove for 30 mins.
Heat the oven to 200C/180C fan/gas 6. Make a gap in the centre of each dough ball, roughly the size of an egg yolk. Spoon the chilled custard into each gap, then brush the exposed dough with the beaten egg. Bake for 18-22 mins until golden and risen. Leave to cool completely.
Mix the icing sugar with 2 tsp water, then brush over the exposed dough. Sprinkle with the desiccated coconut and leave to set slightly before serving.