Mix the yeast with half a teaspoon of the sugar and two tablespoons of the warm milk. Place in a warm place to rest for 15 minutes, or until frothy.
Sift the flour and salt into a large bowl. Stir in one tablespoon of the sugar.
Make a well in the flour and pour in the yeast mix, the rest of the milk, the melted butter and the egg. Mix to make a dough and then knead, cover the bowl and leave to stand for 45 minutes, or until the dough has doubled in size.
On a floured work surface, knead the dough for five minutes, then divide into 12 balls and place in a warm place for 30 minutes, or until they have doubled in size.
Heat the oil in a deep pan until it reaches 190C/375F, or a cube of bread dropped in sizzles and turns golden in 30 seconds. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
Gently lower the dough balls one at a time into the hot oil, in batches of two or three, and fry for 3-5 minutes or until golden-brown, and then carefully turn over. Remove from the hot oil with a slotted spoon and drain on kitchen paper.
Roll the hot doughnuts in the remaining sugar. Make a small slit in the doughnut with the tip of a knife, and fill with a teaspoon of jam. Serve.