Dissolve the yeast and sugar in the warm water. Let solution stand for 5 minutes or until it becomes foamy on top. Make sure the water isn't too hot to avoid killing the yeast.
Sift flour and salt together in a large bowl. Beat egg yolk in a small bowl, then stir in the milk and vanilla. Pour the yeast solution into the bowl with the egg and milk, and stir. Make a well in the center of he flour, then pour the yeast solutioninto that well. Stir by hand gently with a wooden spoon from the middle, slowly bringing in more flour until all ingredients come together into a ball-the dough will be too sticky to handle at this point. Cover the bowl and let sit in a warm place for 30 to 45 minutes or until the dough doubles in size.
While the dough is rising, cut ten 4 x 4-inch squares of wax paper and arrange them on two rimmed baking sheets. Each doughnut will rise on its own square of wax paper, and the rimmed baking sheet will allow you to wrap a large sheet of foil over the doughnuts without the foil touching the doughnuts. Sprinkle a little flour on each of the squares of wax paper.
Use flour on your hands to transfer the dough to a floured surface. Sprinkle a little flour on top of the dough, and then gently roll out the dough until until it's about 1/2 inch thick. Use only the amount of flour needed to allow you to handle the dough. Use a 3-inch biscuit cutter to cut out circles of dough, then use a lid from a plastic soda or water bottle (about 1 1/8-inch diameter) to cut the holes. Carefully place each doughnut on a square of wax paper with the floured side of the doughnut down on the wax paper. Cover the doughnuts with large sheets of foil and let them sit 45 minutes in a warm place until the doughnuts have puffed up to at least twice their original size.
While the doughnuts rest, make the glaze by combining water and granulated sugar in a small saucepan over medium heat. Bring mixture to a boil and then reduce heat and simmer for 1 minute, or until the solution is clear. Pour hot sugar solution into a medium mixing bowl, add the powdered sugar, meringue powder, and a pinch of salt, then use an electric mixer and mix for 2 minutes on high speed. Cover bowl with plastic wrap until you need it.
As the doughnuts rise, heat vegetable shortening in a fryer to 375 degrees F.
When the doughnuts have doubled in size, carefully transfer 2 or 3 doughnuts at a time to the hot shortening. Turn each doughnut over onto a palm of your hand, carefully remove wax paper, and then drop it into the oil. Fry for 1 minute, flip doughnut with the handle of a wooden spoon, and continue frying for an additional minute. Remove doughnuts to a cooking rack for one minute.
After a minute on the cooking rack, spoon the glaze over the top of each doughnut so that it runs down the sides and off onto the plate below the cooking rack. In between each glazing stir the glaze so that it stays smooth. If glaze begins to thicken, stir in just a little bit of hot water to thin it out. Let the glazed doughnuts cool 5 minutes before eating. Don't store doughnuts in a sealed container or moisture from the doughnuts will melt the glaze. Instead, store them covered loosely with aluminum foil. Stored this way the doughnuts will be good for about 1 day