To make the filling, place the apples, sugar and just enough water to moisten in a medium pot over low to medium heat. Cook for 25 minutes until the apple appears slightly translucent.
To make the dough, place the milk, sugar and eggs in a bowl and whisk with a fork. Using a KitchenAid or similar mixer, add the flour, salt and yeast. Using the dough hook attachment on low speed, combine the dry ingredients, then slowly add the milk mixture. Mix for 5 minutes. Remove bowl, cover with clingfilm and place bowl over hot water to rise for 40 minutes.
Add the butter and knead for 5 minutes then cover and leave to rise in a warm spot for 1 hour 30 minutes.
Tip dough onto a lightly floured bench and roll to 1/2 cm thick. Using two round cutters, one 5 cm diameter and one 4 cm diameter, cut out the same number of each round, making as many as you can by re-rolling the scraps. The 5 cm rounds are the top of the doughnuts, the 4 cm shapes are the bases.
Whisk the extra egg to make an egg wash. Brush the 4 cm rounds and place a teaspoon of apple onto the round. Top with the 5 cm round and pinch together all the way around.
Repeat for all rounds, placing doughnuts on a tray lined with nonstick paper, leaving 2 cm between each one. Cover the doughnuts with a damp teatowel and allow them to rise for 20 minutes in a warm place.
Preheat a deep-fryer to 170°C and drop a couple of the doughnuts in at a time. Fry till light golden on each side, then transfer to a cooling rack. It’s possible to use a saucepan, but you need to use a thermometer to ensure the oil stays around 170°C.
To make the icing, place all ingredients in a medium bowl and mix to make a honey consistency. Dip the top half of the doughnuts in the icing and allow to set. Serve and enjoy the same day.