• Pour warm water into a small bowl and whisk in yeast and 1 teaspoon sugar. Let stand until a foamy layer forms on top, about 5 minutes.
• Melt shortening with milk in a small saucepan over medium-low heat until tiny bubbles appear on surface. Transfer to a large bowl along with salt and remaining 1/4 cup sugar, and beat with an electric mixer at low speed just until sugar dissolves.
• Add 1 cup flour and eggs and beat at low speed until smooth. Beat in yeast mixture and remaining 3 cups flour until combined. Continue beating until dough pulls away from sides of bowl, about 2 minutes.
• Turn dough out onto a well-floured surface and knead until smooth and elastic, about 2 minutes.
• Shape into a ball and transfer to a large greased bowl. Turn dough to grease evenly, cover with plastic wrap and let rise in a warm place until doubled, about 50 minutes.
• Punch down dough. Turn out onto a lightly floured surface and roll out 1/2 inch thick. Cut out donuts using a floured 3-inch donut cutter. Loosely cover donuts with wax paper and let rise until very light and fluffy, about 40 minutes.
• Pour oil to a depth of 2 inches in a deep, heavy-bottomed skillet or pot (preferable cast iron) or a deep fryer; heat to 375 F.
• Working in batches of 2 or 3 at a time, gently pick up and drop donut rings into hot oil. Fry, turning once, until light golden brown, about 2 minutes total.
• Transfer with a slotted spoon to a baking sheet lined with paper towels or brown paper to drain.

Making the Glaze:

• Stir together sugar, vanilla and enough water to make a thin glaze. Dip donuts in glaze or spread a thin layer over top of warm donuts.