Pour 1/2 cup of warm water into a small bowl, and stir in the yeast and 1 teaspoon of sugar. Allow the yeast mixture to stand until it forms a creamy foam layer on top, 5 to 10 minutes.
Place milk and vegetable shortening into a small saucepan over medium-low heat and cook until the shortening has melted and the milk has tiny bubbles around the edge. Remove from heat.
Place 1/4 cup of sugar and the salt into the work bowl of a large stand mixer and pour the milk and shortening into the bowl. Mix on low speed just to dissolve sugar; and allow mixture to stand until warm.
Beat in the yeast mixture, eggs, and 1 cup of flour on low speed until smooth. Gradually beat in remaining flour to make a soft dough.
Continue beating the dough on low speed until dough leaves the sides of the bowl, 2 to 3 minutes.
Turn the dough out onto a well-floured work surface and knead until smooth and elastic, 2 or 3 more minutes.
Form the dough into a ball and place into a large greased bowl. Grease the top of the dough, and cover the bowl with waxed paper. Set the bowl into a warm place; let the dough rise until doubled in size, 1 to 1 1/2 hours.
Punch down the dough and roll out on a floured work surface to 1/2 inch thick. Cut doughnuts out of the dough with a 3 1/2-inch round doughnut cutter with a 1-inch smaller cutter in the center. Let the doughnuts rise on the work surface until very light and fluffy, 40 minutes to 1 hour.
Heat oil in a deep-fryer or large saucepan to 350 degrees F.
Gently pick up and drop the doughnuts into the hot oil, working in small batches of 2 or 3 at a time, and cook until the doughnuts are light golden brown, turning once, about 2 minutes.
Drain the doughnuts on paper towels to absorb oil.
In a bowl, mix 1 tablespoon of warm water and vanilla extract, and stir in the confectioners' sugar to make a glaze. Spread a thin layer of glaze over the warm doughnuts.
