Step 1
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.
Step 2
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.
Step 3
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.
Step 4
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.
Step 5
Continue beating the dough on low speed until dough leaves the sides of the bowl 2 to 3 minutes.
Step 6
Turn the dough out onto a well-floured work surface and knead until smooth and elastic 2 or 3 more minutes.
Step 7
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.
Step 8
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.
Step 9
Heat oil in a deep-fryer or large saucepan to 350 degrees F (175 degrees C).
Step 9
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.