Sift the flour into a large bowl, add the salt, sugar and yeast and mix together. Place the butter, milk and vanilla extract together into a small pan and warm over a very gentle heat until the butter has melted and the milk is just warm but not boiling (you should be able to put your finger in it without burning it).
Stir in the egg. Make a well in the middle of the dry mix and gradually add the milk mixture and stir with a wooden spoon to form a rough dough. Tip out onto a floured surface and knead for 10 minutes adding more flour as necessary or until the dough is not sticky and slightly springy to touch. Place back into a clean, lightly oiled bowl, cover with a piece of greased cling film and leave to rise in a warm place for 1 ½ hours or until doubled in volume.
Punch down the dough with your fist, knead lightly then divide into 12 equal pieces. Roll them between your palms to form balls and then place on baking sheets well spaced apart. Cover with a piece of greased cling film and leave to rise again 45 minutes (check time) or until doubled in size. (This will make the doughnuts light and fluffy once cooked).
Roll over the top of the doughnuts to make them approx. 3cm in height – otherwise they will be huge! – and then using a lightly oiled 4cm pastry cutter, stamp out the middle of each doughnut and set aside. You can use these to make mini jam doughnuts later.
Pour the oil into a large saucepan to the depth of 10cm and heat to 180-190C or when a small piece of dough dropped into it sizzles immediately and floats to the surface. Carefully lower 2 or 3 doughnuts at a time on a slotted spoon and fry for 30 seconds on each side or until golden brown.
Remove with a slotted spoon and drain on kitchen paper then roll in the cinnamon sugar to coat. Keep warm in a low oven. Repeat while you fry the rest of the doughnuts and then serve warm.
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