In a standing mixer, combine lukewarm water and yeast. Let it sit until dissolve for 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.
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 large 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.
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 prepared paper towels.