Start the Dough: In the bowl of your stand mixer, combine the yeast, sugar, shortening, and warm milk. Mix with the dough hook for about 2 minutes until well blended.
Add Wet & Dry: Stir in the beaten egg. Then, gradually add 4 cups of the flour and the pinch of salt. Mix on low speed until all the flour is just incorporated, which takes about 1 minute.
Form the Ball: Increase the mixer speed to medium and continue mixing until the dough forms a soft ball that pulls away from the sides of the bowl and climbs up the dough hook. Remember, the dough should be soft, not stiff!
First Rise: Gently remove the dough from the bowl, form it into a smooth ball, and place it in a lightly oiled bowl. Turn the dough to lightly coat all sides. Cover the bowl with plastic wrap and let it rise in a warm, draft-free place for about 2 hours, or until it has doubled in size.
Prep for Frying: While your dough is finishing its rise, get your frying oil ready. Pour enough oil into a deep pot or fryer and heat it to 360°F (180°C).
Shape the Beignets: Generously flour your work surface, hands, and rolling pin. Turn the risen dough out and gently pat it into a rectangle about 1-inch thick. Lightly dust the surface of the dough with flour. Roll the dough out to a rectangle roughly 12.5 inches long by 10 inches wide and about ¼-inch thick. Use a sharp, floured knife or dough cutter to cut the dough into twenty 2.5-inch squares. Avoid re-rolling any leftover dough scraps, as this can make your beignets tough.
Fry to Perfection: Carefully place a few beignet squares into the hot oil at a time, making sure not to overcrowd the pot. Fry for about 3 to 5 minutes, turning them until they are beautifully golden brown and crispy on all sides. Overcrowding can lower the oil temperature and prevent them from puffing up!
Serve Sweet: Remove the fried beignets and drain them on a plate lined with paper towels. While still warm, generously dust them with powdered sugar. Serve immediately and enjoy these delightful, pillowy treats!