In a bowl of a standing mixer, sprinkle the yeast over the warm water and stir to dissolve. Then add the sugar, eggs, and milk and blend with a paddle attachment. Add half the flour, salt, and shortening and blend well. Add the remaining flour and continue mixing to form a dough. Place the dough on a flour work surface and form into a ball. Place the dough in a greased bowl, and cover with plastic wrap then refrigerate overnight.
The next day, punch down the dough on a work surface, then cover it with a damp towel, and let it rest for 15 minutes. Then roll the dough into a rectangle 1/4-inch thick. With a pizza cutter, cut into 2-inch squares.
Heat some oil for deep fat frying to 365 degrees F. Drop in the beignets, 3 or 4 at a time, and fry on both sides until golden brown. They will look like pillows. Drain them on a brown paper bag and dust extremely heavily with powdered sugar.
These are sold everywhere in New Orleans, even at the airport with the powdered sugar in their white oil stained paper sack or the option of a bag of powdered sugar separate. Probably the most famous place for these is Cafe Du Monde in New Orleans. I went there this year and thought 2 would be enough, but they're so light I was still craving them after the third one. You eat them traditionally with steaming cups of cafe au lait and the coffee is brewed with chickory sometimes called Creole Coffee. Coffee is brewed with roasted chickory, the root of the leafy salad green, to extend and give body and aroma to the coffee. To make cafe au lait, warm milk in a saucepan and pour 1/3 milk and 2/3 coffee into your cup. Some places throw their beignet dough scraps into the mix for the next batch of dough, almost like a sour dough starter or levand.