Mix the flour, salt, yeast and sugar in a food processor, add the soft butter, egg, yoghurt and 125ml of lukewarm water and pulse until smooth. Tip onto a lightly-floured surface and knead until smooth. If necessary add a little more flour, then roll into a ball.
Clean out the bowl and lightly oil it. Place the dough inside and cover with cling film and put in the fridge overnight (the dough can be kept up to 48 hours in the fridge).
To make the apple purée and the filling cut the apple quarters into 5mm slices across and place straight into the pan, add the sugar and apple juice. Cover and cook on a moderate heat for about 7 minutes until just softening.
Drain over a bowl, keeping the juice, and roughly chop the apple pieces before allowing to cool. Put 100g of the fruit to one side and reserve for the filling. Pour the rest into a food processor with the juice and blitz to a textured purée.
Cut the dough into 4 and work with a piece at a time, keeping the remainder covered as you go. Roll out on a lightly floured surface to about 5mm thickness or even thinner and stamp out 7 circles from each piece with a 5.75cm round cutter.
Put a quarter teaspoon of cooked apple in the centre of each circle, dab a little water around the edges and pinch together to seal well. Spread the dough balls across two baking trays dusted with flour and cover with dry cloths. Leave for about 30 minutes-1 hour until they almost double in size.
Heat the oil to 175°C and fry the doughnuts in small batches for about 3-5 minutes until golden, turning them as they cook. Toss in sugar while still hot.