In a small bowl, combine the warm water (110 degrees) and 1 tablespoon of sugar. Sprinkle the active dry yeast over the top and let it sit for about 5-10 minutes until it becomes frothy.
In a large mixing bowl, combine the warmed evaporated milk, the remaining sugar, salt, vegetable shortening, egg, egg yolk, and vanilla extract. Mix well until everything is nicely blended.
Once the yeast is activated, add it to the mixture of wet ingredients. Stir until combined.
Gradually add the all-purpose flour to the mixture, starting with 2 1/2 cups. Stir until a soft dough forms. If the dough is too sticky, add more flour, a little at a time, until it reaches a workable consistency.
Transfer the dough to a lightly floured surface and knead for about 5-7 minutes until it’s smooth and elastic.
Place the dough in a greased bowl, cover it with a clean kitchen towel, and allow it to rise in a warm place for about 1-2 hours or until it has doubled in size.
Once risen, punch down the dough to release the air. Roll it out to about 1/2 inch thickness. Using a doughnut cutter, cut out your doughnuts and doughnut holes.
Place the shaped doughnuts on a parchment-lined baking sheet, cover them, and let them rise again for about 30-45 minutes until puffy.
In a deep fryer or large pot, heat the vegetable shortening to 350 degrees Fahrenheit.
Carefully drop the doughnuts into the hot oil, frying a few at a time. Fry for about 1-2 minutes on each side until golden brown. Use a slotted spoon to transfer them to 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 smooth and creamy.
Dip the warm doughnuts into the glaze, allowing the excess to drip off. Place them back on the cooling rack to set.
