In large bowl, dissolve sugar and yeast in warm water; let stand until frothy, about 10 minutes. Stir in flour until you have a smooth paste. Cover with plastic wrap; let stand until bubbly and stretchy, about 30 minutes.
In large bowl, beat butter with sugar until fluffy. Beat in eggs, vanilla and salt, beat in sponge. Stir in flour until you have a sticky dough. Turn out onto a lightly floured surface and knead for 6-8 minutes, or until smooth and elastic. Place back in the bowl, cover with plastic wrap, and place in a warm place until tripled in bulk, about an hour and a half.
Punch the dough down and roll or pat it out about 1/2-inch thick. Using a 3-inch doughnut cutter or a round cutter and a smaller one for the hole, cut out doughnuts. Transfer doughnuts and holes to a baking sheet, cover and let rise for another hour, until doubled in bulk.
In a wide, shallow pot, heat 2 inches of oil until hot but not smoking. Cook a few doughnuts at a time for a couple minutes per side, turning as they turn a rich golden. Remove from the pot with a slotted spoon and set aside to drain on a wire rack or paper towels. Makes 1 1/2-2 dozen doughnuts.
Vanilla Icing: Whisk together all the ingredients, adding a little extra icing sugar or milk if needed to make a dippable/drizzlable consistency. Dip or drizzle over doughnuts while they're still warm.