In a small bowl, combine the lukewarm water, yeast, and 1 tablespoon sugar. Set aside to proof until frothy and bubbly, about 10 minutes.
To the bowl of a stand mixer fitted with the dough hook, add the yeast mixture, 2 cups flour, the remaining 1/3 cup sugar, the beaten egg, butter, evaporated milk, vanilla, and salt. Mix on low speed until combined. Add 1-1/2 cups flour, 1/2 cup at a time, and continue mixing on low speed until a soft dough forms, 5 to 6 minutes. If the dough is still too sticky to work with, continue adding 2 tablespoons flour at a time, up to an additional 1/2 cup.
Lightly oil a large bowl. Form the beignet dough into a ball and add to the bowl, cover with plastic wrap, and let rise in a warm, dark place until doubled in size, at least 1 hour.
On a lightly floured work surface, roll the dough out about 1/3 inch thick. Cut into 2 x 2-inch squares.
Fill a Dutch oven or deep pot with enough oil to cover by 3 inches and heat to 370°F. Working in batches of two or three, fry the beignets until golden and puffy, 1 to 2 minutes on each side.
Transfer the beignets to a paper towel–lined plate and dust with a generous amount of powdered sugar. Serve warm.
Recipe reprinted with permission from Just Eat by Jessie James Decker (Harper Collins, 2023). Photography by Liz Schnoch.