Here's how I made yeast raised doughnuts. This recipe is excellent and very much like ones you'd buy in a bakery. They are light as a feather! I found this recipe on a blog called Hopefulthings. I made mine in a bread machine but if you'd like instructions for mixing them by hand, I am providing a link to Hope's blog. Be sure to check her out!
Hopefulthings:
http://hopeful-things.blogspot.com/2013/01/homemade-doughnuts.html
Yeast Raised Doughnuts
1 1/4 cups warm water
1 1/4 tbsp. dry yeast
1 tsp. sugar
1/3 cup sugar
1/4 cup shortening
1 egg, beaten
1/2 tsp. salt
1 tsp. vanilla extract
1/4 tsp. nutmeg
3 1/4 cups all purpose flour
shortening for frying the doughnuts in
Glaze:
2 cups powdered sugar
1/2 tbsp. cornstarch
1/4 cup water
1/2 tsp. vanilla
Add 1 cup of the warm water to the bread machine, reserving 1/4 cup to start the yeast in. Add the yeast and 1 tsp. of sugar to the reserved water. Stir to mix well and set aside. Add the rest of the ingredients to the bread machine and when the yeast has proofed, add it as well. Set the bread machine on the dough cycle. When the dough is ready, turn out onto a floured surface and punch down. Let rest for 10 minutes. Roll out to 1/2" thickness and cut out doughnuts. Place on a very lightly greased and floured baking pan turned upside down. Lightly cover with plastic wrap and let rise for about 30 minutes or until double in size. Mix together the glaze ingredients and set aside. Very carefully remove the doughnuts from the baking pan and drop into a pot or deep fryer full of shortening which has been brought to a temperature of 350 to 360 degrees. Fry until lightly brown then flip the doughnut over and fry on the other side. This takes about 1 minute on each side. Remove the doughnuts from the oil and place on a paper towel to drain. Dip in glaze while still warm. Makes about 3 dozen, and this can vary depending on the size of your doughnut cutter, whether or not your doughnuts have holes in them and the thickness you cut the doughnuts.
Note: I would not substitute anything else for the shortening in this recipe. You are not likely to get the same results.