To make the dough, heat the milk until it is just boiling then set aside to cool until it is lukewarm (it is very important that it is not too hot or it may kill the yeast); scalding the milk makes the finished dough softer. Once the milk has cooled, add the yeast, whisk to combine and leave in a warm place for about 5 minutes until foaming and active.
Place the rest of the dough ingredients an a large bowl, add the yeasty milk and mix until it comes together into a dough and you have picked up all the flour from the sides of the bowl. Turn out onto an un-floured surface and knead for 10 minutes or so until the dough is very smooth and elastic. It is a soft dough, but due to the butter and glycerin it will not remain sticky for long once you start kneading.
Place the dough in a lightly oiled bowl, cover with clingfilm and set aside in a warm place for a couple of hours until puffy; it doesn't rise quite as much as standard bread dough.
While the dough is rising prepare the apple filling. Peel, core and dice the apples and place in a large pan with the sugar, water and cinnamon. Bring up to the boil then simmer for about 10 minutes, stirring regularly, until the apples have broken down and become mushy. Add a splash more water if needed though bear in mind that you need the sauce to be quite thick.
Puree the apple sauce using a stick blender (any chunks of apple make it difficult to pipe into the doughnuts). Taste and add more sugar and cinnamon as needed; cover and refrigerate.
Give the risen dough a brief kneading to knock it back then roll it out to about 1cm thick (flour the surface a little if needed). Use a 6-7cm round cutter to cut out as many circles as possible and place them well-spaced apart on a baking parchment lined tray. Re-roll the scraps and cut out more circles.
Loosely cover the doughnuts with lightly oiled clingfilm and set aside to rise until puffy, about 45 minutes.
While the doughnuts are rising, mix together the sugar and spices in a shallow bowl. Heat a deep fat fryer or large pan of oil to 180C/350F.
Fry the risen doughnuts a few at a time for about 2-3 minutes per side until deep golden; when cooked, a skewer inserted into the centre of the doughnut should come out completely clean. Drain on kitchen paper then as soon as they are cool enough to handle, toss them in the spiced sugar.
Pierce the side of each doughnut with a skewer or chopstick and wiggle it around a bit to create space inside the doughnut. Place the apple compote in a piping gun, syringe or cookie press fitted with a medium round nozzle (you can use a piping bag but it it far easier to use a gun/syringe). Fill each doughnut with apple sauce until it starts to ooze out.