For the dough, warm the milk and put 50ml/1¾fl of it into a small bowl with the yeast and half a teaspoon of sugar. Set aside for 15 minutes to allow the yeast to reconstitute.
Mix the flour, salt and two tablespoons of sugar in a freestanding mixer with a dough hook attachment. Add the yeast mixture, the remaining warmed milk, 100ml/3½oz water, the butter and eggs.
Mix with a dough hook for 10 minutes, or until the dough is smooth and elastic. Then cover and set aside to prove for 45 minutes.
Meanwhile make the apple filling. Put all the filling ingredients in a saucepan set over a medium heat. Add four tablespoons water and cook until the apples are mushy. Set aside to cool.
For the salted caramel, melt the sugar in a pan until it reaches a deep-brown colour. Then remove it from the heat and immediately whisk in the butter.
Add the cream slowly, whisking until combined. Mix in the salt and then pour the caramel into a bowl and set aside.
For the dough, briefly knead the risen dough on a floured work surface to knock it back.
Divide the dough into 12 equally-sized balls then roll into neat spheres.
Grease a large baking tray with a little butter. Place the dough balls on the tray, cover with a floured tea towel and leave to rise until doubled in size (about an hour).
To cook the doughnuts, preheat a deep-fat fryer to 190C/375F, or alternatively heat the oil in a large heavy-based pan until a piece of bread sizzles and turns golden-brown when dropped into it. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
Fry the dough balls in batches (do not crowd the pan), turning occasionally to ensure they are an even colour. Cook for around 3–5 minutes, or until golden-brown.
Once cooked, carefully remove the cooked doughnuts using a slotted spoon. Drain on kitchen roll and, when dry, set aside to cool on a wire rack.
To assemble the doughnuts, mix the cinnamon and caster sugar together in a bowl.
Blend the apple filling in a food processor, or using a hand-held blender, until you have a smooth purée.
Fill a large piping bag with the apple purée. Alternatively spoon the purée into a piping syringe with an injector nozzle attached.
When the doughnuts are completely cool, use a small, sharp knife to make a cut into the side of each doughnut, reaching to the centre. Pipe, or syringe, the apple purée generously inside.
Dip one half of the filled doughnuts in the cinnamon sugar. Dip the other side into the cooled salted caramel. If the caramel is too hard, heat it very gently to loosen, but do not allow it to get so hot that it can’t be handled safely.
Set the doughnuts aside to allow the caramel to settle before serving.