Warm the milk in a saucepan. Add the butter and set aside until the milk has cooled to hand temperature and the butter has melted. Put the flour in a large bowl with the yeast, sugar, cinnamon and 1?2 tsp salt, mix well. Make a well in the centre and pour in the warm milk mixture, egg and apple. Combine with a wooden spoon, then tip out onto a work surface and knead for a few mins to combine. Pop into a lightly oiled bowl, cover with oiled cling film and leave somewhere warm to rise until doubled in size – about 2 hrs.
Lightly grease 2 large baking trays. Uncover the dough and knock out all the air. Remove a lump of dough, roughly the size of a walnut, and roll into a smooth ball. Put on a tray and squash gently with your palm. Repeat with the remaining dough – you should make about 20 doughnuts in total. Cover the trays with a sheet of oiled cling film and leave to prove until doubled in size again – about 30 mins.
Line a large plate or baking tray with kitchen paper, and mix the remaining sugar and cinnamon on another. Pour enough oil to come halfway up the sides of a large saucepan. If you have a thermometer the temperature should reach 180C. If you don’t have one, drop
in a small chunk of bread. The oil is ready when it browns in about 30 secs. Drop in 3-4 doughnuts at a time (depending on the size of your pan) and cook for 4-5 mins until each one is deep golden brown and puffed up. Drain on the kitchen paper, then quickly toss in the cinnamon sugar mixture. Repeat with the remaining doughnuts.
To fill the doughnuts, use a skewer or cocktail stick to make a hole in each one. Wiggle it around in the middle to create a cavity. Transfer the caramel to a piping bag fitted with small nozzle, insert the nozzle into the doughnut and squeeze as you pull it out, filling generously. Continue with the remaining doughnuts. Serve with extra caramel for dunking.