In a small bowl, combine the flaxseed and water with a spoon. Set it aside and wait at least 5 minutes for it to thicken. It will be similar to a jelly, once it's ready.
Once the flax egg is ready, add the apple sauce and mix well.
Warm up the milk of your choice until it's just about warm (not too hot!) and add the yeast. Let it stand for at least 5 minutes, or until it's foamy and bubbly.
Once the yeast and milk mix is bubbly, add the apple cider vinegar and stir well to incorporate.
Now get your standing mixer ready with the dough hook and to the bowl add the dry ingredients, the yeast mixture, the grated apple and the rest of the wet ingredients. Mix them until they are well combined and incorporated.
Knead the dough until it becomes elastic and glossy, and it clings to the dough hook. This can take about 10 to 15 minutes.
Place the dough in a lightly greased bowl and cover it with cling film.
Let it rise in a warm spot for about 1 hour or so, until it doubles in size.
Once it's reached the right size, lightly coat a clean working surface with flour and deflate the dough - punch it real good! Transfer the dough to the worktop and roll it out to a 2.5cm thickness.
Cut the shape of the doughnuts by using a cookie cutter or a doughnut cutter. We used a cookie cutter and found it easier to create the middle hole using our fingers.
Once cut, place each doughnut on a tray covered with parchment paper and let it rise for a further 30 to 40 minutes.
Once ready to fry, warm up the oil to 180°C using a thermometer and fry each donut and donut hole for a few minutes, until golden on each side. Let them dry out on some kitchen roll-covered plate for a couple of minutes.
Mix the caster sugar and cinnamon, then roll each doughnut and doughnut hole and set aside to serve.