In the bowl of a standing mixer, combine lukewarm water and yeast. Allow it to sit for about 5 minutes until the yeast dissolves. Meanwhile, heat the milk in a microwave-safe medium bowl for about 2 minutes. Once heated, remove it and allow it to cool. Add the cooled milk, sugar, salt, eggs, shortening or butter, and 2 cups of flour to the yeast mixture. Mix at medium speed for 2 minutes. Gradually add the remaining 3 cups of flour and continue mixing, scraping down the sides of the bowl as needed.

Transfer the dough to a large greased bowl, cover it loosely with a clean cloth, and let it rise in a warm, draft-free place for 1 to 2 hours, or until doubled in size. Once the dough has risen, roll it out on a floured surface to about ¼ inch thick. Using a doughnut cutter or two round cookie cutters (one about 1 inch and one about 3-4 inches), cut out doughnut shapes. Let the doughnuts stand for about 10 minutes.

In a large saucepan, pour vegetable oil to a height of at least 3 inches (about 5 centimeters). Heat the oil over medium heat until it reaches 375°F (190°C). Carefully drop a few doughnuts at a time into the hot oil, frying them for about 3 minutes, turning once, or until they are golden brown. Remove the doughnuts from the oil and drain them on paper towels.

Doughnut Glaze

In a microwave-safe bowl, melt the butter. Once melted, stir in the powdered sugar and vanilla extract until smooth. Gradually add evaporated milk (or water as a substitute) until the glaze reaches your desired consistency. Dip the fried doughnuts into the glaze, allowing the excess to drip off before placing them on a rack to set.