Combine the yeast and warm water in the bowl of a stand mixer fitted with a dough hook. Let stand for 10 minutes.

Add the sugar, salt, butter, egg, egg yolk, vanilla extract, and 2½ cups of the flour. Knead on low speed until a dough begins to form. If the dough is sticky, add more flour a tablespoon at a time until a soft, tacky dough forms.

Transfer the dough to a lightly oiled bowl, cover tightly with plastic wrap, and place in a warm, draft-free spot for 2 hours.

Gently press to deflate the dough, wrap in plastic wrap, and refrigerate for at least 8 hours, or overnight.

On a floured work surface, roll the dough out to a ½-inch thickness. Cut out the doughnuts and place on a parchment-lined baking sheet. Cover and allow to rest for 1 hour.

Line a large baking sheet with paper towels and place a wire cooling rack on top.

Heat the vegetable shortening in a large skillet to a maintained temperature of 360 to 370 degrees. Gently lower the doughnuts into the oil, cooking until golden brown, about 1 minute per side.

Remove the doughnuts and place on the cooling rack. Repeat until all doughnuts are fried.

In a medium bowl, whisk together the powdered sugar, water, and vanilla extract until smooth.

Dip each doughnut into the glaze and return to the cooling rack. Allow the glaze to set for about 15 minutes, then serve.