Place the flour in a bowl, with the sugar and yeast on one side and the salt on the other. Mix to combine.
Put the egg, warm milk and butter in a jug and mix well. Make a well in the centre of the dry ingredients and pour in the liquid. Mix with a spoon, then bring it together with your hands into a soft dough.
Lightly grease a clean work surface with vegetable oil and knead the dough for 10 minutes. The dough is ready when it is smooth, shiny and elastic.
Grease the inside of a large bowl with a little oil, put the dough in, and cover with cling film. Leave in a warm place to double in size. This can take about an hour, sometimes more, sometimes less – it really does depend on the warmth of the room.
Once doubled in size, knead the dough for a minute to knock the air out. Divide into 12 equal pieces and shape into balls. Place on a greased baking tray and cover with a piece of oiled cling film. Leave for 30 minutes to 1 hour until the dough doubles in size and springs back when you press with your finger.
Fill a deep-fat fryer or a large, deep, heavy-bottomed pan two-thirds full with the oil. Heat to 190C – a cube of bread should turn brown in one minute when dropped in. (CAUTION: Hot oil can be dangerous. Do not leave unattended.)
Fry the doughnuts in batches, cooking for 3 minutes on each side, moving them around in the oil. Drain briefly on kitchen paper, then toss them in the vanilla sugar. Use a knife to make a small cut through the side and wiggle it around to make a hole in the centre. Carefully pipe in the jam, being careful not to overfill. Delicious.