Whisk the warm water with ⅓ of the sugar and add the active yeast. Whisk again and leave to stand for 5 minutes.
Add the remaining sugar, buttermilk, eggs and melted butter to the yeast solution. Whisk until the sugar dissolves.
Mix the liquids with the flour and knead the dough until smooth, about 10 minutes. Transfer to an oiled bowl, cover and leave to rest for 1 hour.
Roll out the dough and cut into circles. Use a small round cookie cutter to cut out half of half of the dough circles.
Sprinkle a little water all over and place the circles with the hole in the centre facing up. Press down lightly. Leave to rest for 15 minutes.
Heat the vegetable oil to 180°C/350°F. Fry the doughnuts on both sides for 3-4 minutes. If the centres puff up, make a small slit to let the air out.
Coat the golden donuts with sugar and fill each hole of the donut with the custard.
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