Dissolve 1 teaspoon of sugar in warm water. Sprinkle yeast over the top. Let stand for about 5 minutes to dissolve.
In a large bowl, whisk the eggs and 1 cup of sugar until sugar has completely dissolved. Stir in the salt and flour. Mix in the yeast mixture, milk and melted butter. A wooden spoon works well for this. Cover and set in a warm place to rise until doubled.
Once the dough has doubled, punch down and knead until the dough is elastic. Cover and allow to rise again. If you like, you may refrigerate overnight. When the dough has doubled, it is ready to use.
Heat oil in a deep fryer or large deep pan to 180 degrees C. Set a small dish of water to the side for wetting your hands slightly before touching the dough. This will help it keep from sticking. Only wet your hands a little, excess water dripping in the hot oil is very dangerous.
With slightly wet hands, gather a small ball of dough about the size of a golf ball. Stretch the dough out a little bit while turning around until you have a disc about 10cm wide. If you mess up, just throw it back in with the other dough and start over. Tiny holes are fine.
Carefully place the stretched dough into the hot oil. The doughnuts should puff up a bit as they hit the oil. Cook until golden brown, about 3 minutes per side. Remove from the oil using a slotted spoon. Drain on paper towels. Coat with extra sugar while still hot. Serve warm.