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Melt the butter in a pan over a low heat. Gently warm the milk in another pan. Mix the yeast with 1 tsp of sugar and 40ml warm milk, and set aside for 15 mins.
Sift the flour and a large pinch of salt into a bowl and stir in the rest of the sugar. Pour in the yeast mixture, the rest of the milk (reheat if you need to, but make sure it’s not too hot), the melted butter and the egg, then mix into a dough. Tip out onto a lightly floured surface and knead for 5-10 mins, then put in a bowl and cover with cling film. Leave for 45 mins-1 hr, or until doubled in size.
Meanwhile, make the jam. Put a saucer in the freezer. Warm the sugar over a low heat in a saucepan for a few mins, being careful not to burn it. Add the blackberries and cook until they release their juice, then add the lemon juice, turn up the heat and bring to the boil, stirring gently.
Skim off any surface scum and boil for about 8 mins. Spoon some jam onto the cold saucer – if it forms a skin that wrinkles when touched, it’s ready. If it isn’t ready yet, keep cooking and testing. Leave to cool slightly while you finish making the doughnuts.
Knock the air back from the dough for a min or so, then divide and shape the dough into 10 balls. Transfer the balls to oiled baking sheets and cover with cling film – make sure it’s puffed up so it doesn’t touch the doughnuts. Leave to prove for 30 mins.
Heat the oil in a deep-fat fryer, or fill a saucepan one-third full of oil and heat until it reaches 180C (a piece of bread should turn golden in a few seconds). Put some sugar on a large plate. Fry the doughnuts in batches of two or three for 1 min each side until golden. Remove with a slotted spoon, drain on kitchen paper and roll immediately in sugar.
Use a jam syringe or piping bag to fill the doughnuts with jam (make a little slit in the top of each doughnut with a knife first if that helps), then dust with more sugar. Serve with crème anglaise or custard. If you have any jam left over, spoon it into a sterilised jar. The jam will keep for a couple of weeks.