Start by activating yeast. Take yeast in a bowl, add in 1 tsp of sugar and 3 to 4 tblspn of warm milk. Leave it aside for 5 mins till the yeast is foamy and bubbles up.
Now take egg, sugar, vanilla,, milk, salt in a large mixing bowl and beat it well so the sugar is melted.
Now add in the foamed yeast mix and mix well.
Add in flour and mix well with a spoon and knead it to form a dough. It will be a soft, non sticky dough. Cover the dough with a cling wrap and leave it to rise for 1 hour (In colder region it may take 2 to 3 hours)
Once it is doubled in size, punch the air out and transfer the dough to a floured work surface. Roll the dough to 1 ot 2 cm thickness. Use a cookie cutter to cut out rounds.
Now place the cutouts in a floured tray and cover again and let it rise for 30 to 45 mins. Use the leftover scraps to keep on rolling.
Now Heat oil for deep frying, Once the oil is hot, drop the donuts and fry till golden on both sides, it will take just couple of mins.
Once fried remove them to a paper towel. Once all the donuts is fried, make cinnamon sugar, mix cinnamon and powdered sugar.
Dip the fried donuts in cinnamon sugar and roll them well.
Now for filling the donuts, take jam or preserve in a microwave safe bowl to warm it slightly. I microwave it for 20 sec.
Now fill a piping bag with the jam. take a donut, make a slit using a knife on the side of the donut. Insert the piping bag and give a good squeeze.
You can roll them on cinnamon sugar second time too.