Put the warm water in a large bowl, sprinkle in the yeast and a couple teaspoons of the sugar and stir until thoroughly dissolved. Let proof for 10 minutes. Add the rest of the sugar and the salt, eggs and milk, and blend well. Mix in 4 cups of the flour and beat until smooth. Add the shortening, then the 3 remaining cups of flour, blending well. Cover and chill up to 24 hours.
Roll out the dough to 1/8-inch-thick. Cut into rectangles 2-1/2 inches by 3-1/2 inches. Heat the oil in a deep fryer to 360 degrees. Drop in the beignets, 3 or 4 at a time, and fry on both sides until golden brown. Drain each batch and place on several layers of paper towels in a warm in (200 degree) oven until you're finished frying. Sprinkle with plenty of confectioners' sugar and serve hot with cafe au lait.
Makes 5 dozen