In a small bowl, combine the warm water and 1 tablespoon of the granulated sugar. Sprinkle the active dry yeast over the top and let it sit for about 5-10 minutes, or until it becomes frothy.
In a large mixing bowl, combine the warmed evaporated milk, the remaining granulated sugar, salt, vegetable shortening, egg, egg yolk, and vanilla extract. Mix well until all the ingredients are incorporated. Once the yeast is ready, add it to the mixture.
Gradually add in the all-purpose flour, mixing until a soft dough starts to form. Knead the dough on a floured surface for about 5-7 minutes, or until it becomes smooth and elastic.
Place the dough in a greased bowl, cover it with a clean cloth, and let it rise in a warm place for about 1-2 hours, or until it has doubled in size.
After the dough has risen, punch it down to release the air. Roll it out to about 1/2 inch thickness on a floured surface. Use a doughnut cutter to cut out your doughnuts and holes.
Cover the shaped doughnuts with a cloth and let them rise again for about 30-45 minutes, until they have puffed up.
While the doughnuts are rising, heat the vegetable shortening in a deep fryer or large pot to 350 degrees F. Use a thermometer to ensure the oil is at the right temperature for frying.
Carefully place a few doughnuts at a time into the hot oil, frying for about 1-2 minutes on each side, or until they are golden brown. Remove them from the oil and let them drain on a cooling rack.
In a medium bowl, combine the melted butter, powdered sugar, pinch of salt, evaporated milk, vanilla extract, and hot water. Whisk until the mixture is smooth and has a pourable consistency.
Once the doughnuts have cooled slightly, dip the tops into the glaze, allowing any excess to drip off. Place them back on the cooling rack to let the glaze set.
