Heat the oil in a large heavy-based saucepan or deep-fat fryer over a high heat until it reaches 180°C. Put the jam and 2 tablespoons water in a small saucepan and bring to the boil over a high heat. Boil, stirring occasionally, for 1 minute until the jam melts. Transfer to small dipping bowls and set aside.
Meanwhile, put the flour, baking powder and 2 tablespoons of the caster sugar in a large bowl and mix well. Add three-quarters of the milk and stir to make a soft dough, adding more milk as needed. Turn out the dough on to a lightly floured work surface and roll it out to 1cm thick. Cut out rounds of dough, using a 4.5cm round biscuit cutter, then cut out the centre holes using a 2.5cm cutter. Gather together and re-roll the trimmings, as necessary, to make 24 mini doughnuts.
Working in batches to avoid overcrowding the pan, fry the doughnuts for 2 minutes until puffed and golden, turning once halfway through. Remove from the oil, using a slotted spoon, and drain on kitchen paper. Return the oil to the correct temperature between batches. Toss the doughnuts in the remaining caster sugar and serve warm with the jam for dipping.