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Put the sultanas in a bowl, pour over the rum and leave to soak for 30 mins. Meanwhile, put the butter in a small pan and heat until melted. Continue to cook until the butter starts to turn a nutty brown colour, then remove from the heat and pour a little of the brown butter into the bases of six ovenproof mini pudding basins, swirling it around the sides. Cut the brioche rolls into thirds, then cut a slit in the side of each (if not already split) and stuff a few sultanas into the slits. Put three brioche pieces, cut-side down, into each basin and pour the rest of the butter over and around the sides.
Heat the oven to 160C/140C fan/gas 3. Bring the milk, cream and vanilla almost to the boil. Whisk the eggs with the sugar in a separate bowl, then slowly whisk the hot milk mixture into the eggs until you have a smooth, thin custard. Divide most of it between the basins – do this slowly so the brioche can soak it up. When the basins are almost full, transfer to a roasting tin and fill it with enough boiling water to come halfway up the sides. Top up the basins with the remaining custard, put the tin in the oven and bake for 45 mins. Remove from the oven and leave to cool for 5 mins.
To make the sauce, sprinkle the sugar over the base of a pan and heat, tilting the pan, until the sugar starts to melt and caramelise. Once the caramel is golden, carefully add the cream and rum with a pinch of salt and melt everything together. Turn out the puddings and pour a little of the sauce over each. Serve with extra cream, if you like.