To make the doughnuts:
In a standing mixer, combine lukewarm water and yeast. Let it sit until yeast dissolves, about 5 minutes.
Meanwhile in a microwave safe medium bowl, heat milk for about 2 minutes. Remove and let it cool.
Add, milk, sugar, salt, eggs, shortening or butter and 2 cups of flour to bowl of yeast.
Mix for 2 minutes at medium speed. Add the remaining 3 cups of flour and continue mixing dough. Scraping down sides.
Place dough in a large greased bowl. Cover loosely with a clean clothe and let rise in a warm, draft-free place for about 1 to 2 hours or until doubled.
Tip: In order for these doughnuts to be airy, they have to double in size. So be patient. Depending on the weather, it might take more than the recommended time.
Roll dough out on a floured surface to about 1/4 inch thickness. Cut into doughnuts using a donut cutter or cookie cutter about one 1-inch and one 3 or 4-inch. Let stand for about 10 minutes.
In a deep sauce pan, pour vegetable oil until it is at least 3 inches (or about 5 centimeters) high and place on medium heat until oil is 375 degrees.
Tip: The oil temperature is key. Itís better to add doughnuts to slightly warmer-than-desired oil, than to oil that's too cool, because the dough usually brings down the oil temperature a few degrees.
Carefully drop doughnuts into hot oil, only a few at a time. Fry, turning once, for about 3 minutes or until golden brown. Drain on a cookie rack for a few minutes and then transfer to paper towels.
To make the glaze:
In a microwave safe bowl melt the butter.
Remove and stir in powdered sugar and vanilla extract until smooth.
Add the evaporated milk (or water), small amounts at a time, until you have reached desired consistency.
Dip doughnuts in glaze and place on the rack to let any excess drip off.
Tip: If you want to sugar the doughnuts, instead of glazing them, do so immediately after you remove the doughnuts from the oil, so the sugar will stick.