Take yeast in a bowl. Add warm milk, mix and set aside for five to ten minutes to activate.
Add one cup flour to milk-yeast mixture and mix well till well combined. Cover with cling film and set aside to prove for thirty minutes.
Break egg in another bowl. Add egg yolks and whisk. Add castor sugar, vanilla essence and salt and whisk well.
Add the egg mixture to flour mixture and whisk well using an electric beater. Add melted butter and mix. Add the remaining flour and knead into a soft dough.
Grease another bowl with some butter, transfer the dough, cover with cling film and set aside to prove till it doubles in the size.
Dust the worktop with some flour, transfer the dough and roll out to make 1½ inch thick sheet.
Cut the sheet into doughnut shapes using a doughnut shaped cookie cutter, place them on a baking tray and set aside to prove till it doubles in size.
Heat sufficient oil in a pan. Deep-fry the proven doughnuts till golden and crisp. Drain on absorbent paper.
Spread some castor sugar on a plate and coat fried doughnuts with it.