Combine yeast and warm water in a small bowl or 1-cup glass measuring cup and set aside 10 minutes or until yeast is dissolved and bubbly. I like to sprinkle about 1/2 teaspoon of the sugar I’ve already measured out over the yeast to encourage it. Meanwhile, combine sugar, butter, egg, egg yolk, and salt in a stand mixer. Beat on medium just until combined. Add dissolved yeast mixture and beat on low speed until combined.
Gradually add flour and beat on medium-low speed until combined. The dough should start to pull away from the sides of the bowl. Add one or two more tablespoons if necessary. It will be a sticky dough.
Scrape the dough down to the bottom of the bowl. Spray cooking spray lightly on the top of the dough or brush lightly with oil. Cover with plastic wrap to prevent drying out and let it rise in a warm place (85°) for 1 to 1 1/2 hours or until double in bulk.
Punch dough down; cover with plastic wrap again, pressing it all around the dough to prevent drying out. Refrigerate at least 8 hours or overnight.
On a floured work surface, roll out the dough to between 1/4 and 1/2-inch thickness.Mine was about an 11-inch circle. Using a doughnut cutter or a 2 1/2-inch circle cookie cutter and a smaller circle cookie cutter, cut out doughnut shapes and place on parchment or wax paper. Cover with a towel and let rest 1 hour Heat shortening or vegetable oil to 360° in a heavy 8 to 12-cup pot or deep cast iron skillet.
Prepare the glaze. Combine powdered sugar, water, and vanilla in a shallow bowl.
Gently lower 3 or 4 of the doughnuts into the hot oil being careful not to splash the hot oil. After about 1 minute or when lightly browned on bottom, flip them with tongs. The doughnut holes will take slightly less time. Drain them on layers of paper towels about 20 seconds while the oil returns to 360°
Immediately drop the doughnuts in the glaze and place on wire racks. Repeat procedure with remaining doughnuts and glaze. SERVE WARM or reheat in microwave on HIGH for 7 to 8 seconds if there’s any around to get cold!