Combine the yeast and warm water in a stand mixer bowl. Let it stand for 10 minutes until the mixture is frothy.

Add sugar, salt, butter, egg, egg yolk, and vanilla extract to the yeast mixture. Gradually add 2½ cups of flour and knead on low speed. If the dough feels sticky, add more flour a tablespoon at a time until it forms a soft ball.

Transfer the dough to an oiled bowl, cover it, and let it rise in a warm place for 2 hours until it doubles in size. Press the dough to deflate, then wrap it in plastic and refrigerate for at least 8 hours or overnight.

Roll out the dough on a floured surface to ½-inch thickness. Cut into doughnut shapes and let them rest for 1 hour to rise slightly.

Heat vegetable shortening to 360-370°F in a skillet. Fry the doughnuts for about 1 minute on each side until golden brown. Remove to a cooling rack.

For the glaze, whisk together powdered sugar, water, and vanilla until smooth, then dip the doughnuts and let them set for 15 minutes.