In a small bowl, combine the warm water and 1 tablespoon of the granulated sugar. Sprinkle the active dry yeast on top and let it sit for about 5-10 minutes, or until it becomes frothy.
In a large mixing bowl, whisk together the warmed evaporated milk, remaining granulated sugar, salt, vegetable shortening, egg, egg yolk, and vanilla extract until well blended.
Once the yeast is activated, add the yeast mixture to the bowl with the wet ingredients. Stir until combined.
Gradually add the all-purpose flour, one cup at a time, mixing until a soft dough forms. You may need to add a little more flour if the dough is too sticky.
Transfer the dough to a floured surface and knead it for about 5-7 minutes, or until it becomes smooth and elastic.
Place the kneaded dough in a lightly greased bowl, cover it with a clean kitchen towel, and let it rise in a warm place for about 1-1.5 hours, or until it has doubled in size.
Once the dough has risen, punch it down gently to release the air. Roll it out on a floured surface to about 1/2-inch thickness. Use a doughnut cutter to cut out the doughnuts, and place them on a floured baking sheet.
Cover the cut doughnuts with a kitchen towel and let them rise again for about 30 minutes, or until they are puffy.
In a deep frying pan or fryer, heat the vegetable shortening to 350°F (175°C). Make sure the oil is hot enough before adding the doughnuts.
Carefully add the doughnuts to the hot oil, frying them for about 1-2 minutes on each side until golden brown. Use a slotted spoon to remove them from the oil and place them on a cooling rack lined with paper towels.
In a medium bowl, whisk together the melted butter, powdered sugar, pinch of salt, evaporated milk, vanilla extract, and hot water until smooth and glossy.
Once the doughnuts are slightly cooled, dip the tops into the glaze, allowing any excess to drip off. Place them back on the cooling rack to let the glaze set.
