Combine the apple, 75g caster sugar and lemon juice for the filling in a large saucepan. Cook over a medium heat for 6-8 minutes, stirring frequently until you have a soft compote. Remove to one side and allow to cool.
Combine the milk, water, melted butter, sugar, flour, salt and yeast in a food mixer fitted with a dough hook. Mix for 10-12 minutes until the dough has come together in an even, slightly sticky ball. Turn out onto a lightly floured work surface and allow to rest for 10 minutes.
Roll the dough out to ½cm in thickness and cut out rounds using a 4-5 cm straight-sided cookie cutter. Arrange half of the rounds of dough on a lightly oiled tray. Spoon ½ a tablespoon of the apple compote into the centre of the rounds and wet the rims using a cold water.
Top with the other rounds of dough and seal the edges well to complete the doughnuts. Cover loosely with clingfilm and leave to rise in a warm place until doubled in size; usually 30-40 minutes.
When ready to cook the doughnuts, heat the oil to 180° C in a heavy-based saucepan. Fry in batches until the doughnuts are golden brown in colour, flipping over to ensure they are evenly coloured; usually 2-3 minutes.
Remove with a slotted spoon and drain on kitchen paper. Arrange on a serving plate and dust with the icing sugar. Serve immediately.