In the bowl of an electric mixer, fitted with a dough hook, add the yeast, sugar, shortening, and milk, mix for 2 minutes. Add the egg. Mix well. Add 4 cups of the flour and salt. Beat at low speed until all of the flour is incorporated, about 1 minute. Then beat at medium speed until the mixture forms a ball, leaves the sides of the bowl, and climbs up the dough hook. Remove the dough from the bowl. Using your hands, form the dough into a smooth ball. Lightly oil a bowl. Place the dough in the bowl and turn it to oil all sides. Cover with plastic wrap and set aside in a warm, draft-free place until it doubles in size, about 2 hours.
Preheat the vegetable oil a deep fat fryer to 360 degrees F.
Turn the dough out onto a floured surface and pat the dough into a rectangle about 1-inch thick. Lightly dust the surface of the dough. Roll out the rectangle to 12 1/2 inches long by 10 inches wide and about 1/4-inch thick. With a sharp knife, cut the dough into twenty 2 1/2-inch square beignets. Fry the beignets, a couple at a time until golden brown and crispy on all sides, about 3 to 5 minutes. Remove and drain on a paper towels. Sprinkle the beignets with powdered sugar and serve. Fill each cup with 1/2 cup of coffee and 1/2 cup of the hot milk. Stir well.
Serve the coffee with the beignets.