In a stand 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, 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.
Preheat oven to 375 degrees F. Bake for about 8 to 10 minutes or until the bottom of the doughnuts are golden brown.
Remove from oven and let it cool for about a minute or two.
Dip doughnuts in glaze and let it drip on the rack.
In a mixing bowl sift the powdered sugar, then add all the other ingredients. Mix until all the ingredients have been fully combined.