Combine 50ml warm milk, yeast, and a half teaspoon of sugar in a bowl for 15 minutes.
Mix flour, salt, remaining sugar, yeast mixture, remaining milk, 100ml water, butter, and eggs in a mixer.
Knead with a dough hook for 10 minutes until elastic, then cover and prove for 45 minutes.
Cook filling ingredients with 4 tablespoons of water in a saucepan until apples are mushy, then cool.
Melt caramel sugar in a pan until deep brown, then whisk in butter off the heat.
Slowly whisk in double cream and salt, then set the caramel aside in a bowl.
Briefly knead the risen dough on a floured surface to knock it back.
Divide the dough into 12 equal spheres.
Place balls on a greased tray, cover, and let rise for 1 hour until doubled in size.
Preheat a deep-fat fryer or pan of oil to 190°C.
Fry dough balls in batches for 3 to 5 minutes until golden brown.
Remove with a slotted spoon, drain on paper towels, and cool on a wire rack.
Mix the assembly cinnamon and caster sugar together in a bowl.
Blend the cooled apple filling into a smooth purée.
Transfer the apple purée into a piping bag with an injector nozzle.
Make a small cut in the side of each cool doughnut and pipe in the filling.
Dip one half of each doughnut in cinnamon sugar and the other in salted caramel.
Let the caramel set before serving.