In the bowl of a stand mixer fitted with a dough hook or a large bowl, sprinkle the yeast over the lukewarm water. Stir briefly and allow to sit until frothy, about 10 minutes.
Mix in the sugar, salt, egg, egg yolk, vanilla extract, and flour to create a soft, smooth dough. If too sticky, slowly add a little more flour.
Lightly oil a large bowl and add the dough, turning to coat. Cover and allow to rest in a warm place until doubled, about 2 hours.
Gently deflate the risen dough, cover again, and refrigerate 8 hours to overnight.
Line two baking sheets with parchment. Remove the dough from the refrigerator. On a lightly floured surface, roll the dough into a 1/2 inch thick circle. Use a dough cutter or 3 1/2 inch circle cutter with a smaller 1 inch circle cutter to cut out the doughnuts and arrange them on the prepared baking sheets about 1 1/2 inches apart. Cover and allow to rest until puffed, about 1 hour.
In a deep saucepan or fryer, heat a couple inches of shortening or vegetable oil to 360˚F.
Gently add the doughnuts, a few at a time to not overcrowd the pan, and cook until golden, about 1 minute per side. Remove to a towel-lined plate and repeat with remaining doughnuts. Allow to cool on a wire rack to room temperature before coating in glaze.