Warm the evaporated milk to 110°F and place in a large electric mixer bowl. Stir in 1 tablespoon sugar and the yeast. Let sit for 5 minutes to activate yeast.
Add the flour, shortening, salt, vanilla, egg and remaining 2 tablespoons sugar to the mixer bowl. Knead with the dough hook attachment for 5 minutes on medium-low speed. Scrape down bowl as needed.
Transfer dough to a greased bowl (dough will be tacky), cover with plastic wrap, and let chill in the refrigerator overnight.
The Next Day
On a lightly floured surface, roll the dough to a 10x14-inch rectangle, about ¼” thick. Use a pizza cutter to portion out 12 rough squares. Lay them on a parchment-lined tray and cover with a towel.
Fill a large pot (like a Dutch oven) with vegetable oil 2 to 3-inches deep and warm to 350°F over medium heat.
Fry the dough in batches for 2 to 3 minutes until golden brown, flipping the pieces over every 20 seconds. Transfer to a wired rack.
Place 2 cups powdered sugar in a large bowl. Generously coat the warm beignets with powdered sugar and serve.