In a small bowl, combine lukewarm water, yeast, and part of the sugar. Set aside to rest for approx. 5 – 10 min.
In a large bowl, combine the remaining sugar and salt. Add lukewarm milk, shortening, eggs, and yeast mixture, and mix well using a wooden spoon or a hand mixer. Add a quarter of the flour and mix until dough is homogenous. Gradually add the remaining flour, then knead on a floured surface for approx. 2 – 3 min. until dough is smooth.
Shape dough into a ball, then place in a greased bowl. Cover with plastic wrap or a kitchen towel, then set aside to proof for approx. 1 hr. until dough has risen and doubled in volume.
Roll out dough on a floured surface to approx. 1-cm/0.5-in. thick. Cut out the doughnuts with circular cookie cutters, and cut out a small circle in the center of each doughnut. Cover doughnuts and proof once again for approx. 1 hr.
Heat oil for frying in a pot to approx. 180°C/350°F and fry each doughnut for 1 min. on each side. Cool down on paper towels.
Combine warm water, vanilla extract, and powdered sugar until they form a thick glaze. Coat the warm doughnuts with the glaze, or dunk them in. If desired, decorate doughnuts with sprinkles, nuts, chocolate, or fill with jam or custard.