Combine the yeast, water, and teaspoon of sugar in a medium bowl. Whisk, and let sit a few minutes until the mixture begins to bloom (puff a bit).
Add the oil, vanilla, egg, and confectioner’s sugar. Whisk vigorously for 1 minute.
Add half the flour. Stir, adding more flour as needed, just until it forms a soft, workable dough (it will be very soft). Sprinkle the remaining flour on a counter or board. Scrape the dough onto the counter, and knead in enough flour until the dough is no longer sticky (but still soft). Divide the dough into 6 equal pieces. Let rest for 5 minutes.
Roll each piece of dough into a 10-inch long rope. Fasten the end of each rope to form a circle. Lay a towel on a baking sheet. Dust with flour. Place the formed doughnuts on the pan, cover with a cloth, and let rest for 1 hour, or, until doubled in size.
When ready to fry, add a bit of sugar to a shallow dish. Heat 2-inches of oil in a small pot until it reaches 375ºF. Cook the doughnuts, one at a time, until deep golden brown all over, about 2 minutes per side, turning just once. Drain the doughnuts on a paper towel-lined plate for 20 seconds, then roll in the sugar (if you let them cool too long, the sugar will not stick). Repeat with the remaining doughnuts. Serve hot or at room temperature, but within a few flavors for best flavor.