Lightly grease and flour 3 baking trays. Measure the flour into a large bowl and stir in the yeast. Rub in the butter with your finger tips until the mixture resembles fine breadcrumbs, then stir in the sugar. Make a well in the centre of the dry ingredients and pour in the eggs, milk and water. Mix to a smooth dough.
Turn out on to a lightly floured work surface and knead for about 5 minutes until the dough is smooth and elastic. Return it to the bowl, cover with oiled clingfilm and leave to rise until doubled in size, about 1 - 1.5 hours in a warm room.
Turn the risen dough out of the bowl and knead to knock out the air until the dough is smooth and elastic once more. Divide into 16 equal pieces and shape each into a ball. Flatten each ball, then place a small teaspoon of jam in the centre of each piece. Gather the edges together over the jam and pinch firmly to seal.
Place well apart on the prepared baking trays then cover with oiled clingfilm and leave to prove for about 30 minutes until they have doubled in size.
Heat 2 cm of oil in a deep fat fryer or heavy saucepan until a cube of bread dropped into the fat browns in 30 seconds. Fry the doughnuts a few at a time, turning them once, until they are golden brown all over. This will take about 5 minutes. Lift out with a slotted spoon and drain well on kitchen paper.
Put sugar and cinnamon in a plastic bag then put a couple doughnuts in at a time and shake until the doughnuts are fully covered. Repeat until all your doughnuts are done.