Proof the yeast. In a small bowl, combine warm water, granulated sugar, and active dry yeast. Let it sit for about 5-10 minutes, or until foamy.
Prepare the dough. In a large mixing bowl, combine the flour and salt. Add the yeast mixture, warm milk, beaten egg, and vanilla extract. Mix until a sticky dough forms.
Knead the dough. Turn the dough out onto a floured surface and knead until smooth and elastic, about 5-7 minutes. Place the dough in a greased bowl, cover with plastic wrap, and let it rise in a warm place until doubled in size, about 1-2 hours.
Shape the Beignets. Punch down the risen dough and roll it out on a floured surface to about 1/4-inch thickness. Using a sharp knife or pizza cutter, cut the dough into squares or rectangles.
Fry the Beignets. In a deep fryer or large pot, heat vegetable oil to 350°F. Carefully drop the Beignets into the hot oil, a few at a time, and fry until golden brown on both sides, about 2-3 minutes per side. Remove them with a slotted spoon and drain on paper towels.
Serve and Enjoy. Dust the warm Beignets generously with powdered sugar and serve immediately. Enjoy them hot and fresh for the best flavor and texture.