Stir the yeast and sugar into the warm milk. Set to one side for 10 minutes and allow the mixture to activate and become frothy.
In a mixing bowl or in a stand mixer fitted with the dough hook, add the flour, pinch of salt, melted butter and beaten egg and mix well for at least 2 minutes to form a soft and sticky dough. Cover with a clean, damp cloth and rest the dough for 10 mins.
Hand-knead the rested dough or mix in the mixer with a dough hook for a further 2–3 minutes, until it becomes smooth and supple. Cover and rest again for 20–30 minutes, or until increased in size by about half.
Tip out the dough onto a lightly floured work surface and roll it out to about 1.5cm thick. Cut out six to eight 8cm circles and place each round on greased baking paper, reserving the offcuts.
Cover the doughnuts with a clean tea towel and allow to prove for 30 minutes–1 hour, until doubled in size.
Fill a deep-sided saucepan two thirds full with oil (do not overfill). Heat it to 180°C, or until a little piece of dough sizzles and rises to the surface and browns immediately.
Working in batches, fry each doughnut for about 1½ minutes on each side, until golden all over, then carefully remove with a slotted spoon and transfer to a wire rack.
Coat with extra sugar while still warm.
Put the apple sauce into a piping bag fitted with a medium, round piping nozzle, or use the corner of a small, sturdy plastic bag with the tip cut off.
When the doughnuts are cool enough to handle, push the piping nozzle (or use the tip of a knife to make a hole so that you can push in the plastic bag) into the centre of each doughnut and fill with the apple sauce, then repeat with the custard.
Put all the doughnuts on a large serving plate and drizzle liberally with the maple syrup to serve