In a large bowl, combine the yeast, milk, egg, white sugar, melted butter, and vanilla extract. Stir well.
Add the salt and gradually add the flour, a third at a time, stirring with each addition.
Turn the dough onto a lightly floured surface and knead for 5 to 7 minutes, until the dough is smooth and elastic. Shape into a ball.
Clean and dry the bowl and add 1 tablespoon of vegetable oil and grease the inside of the bowl. Place the dough ball in a bowl and cover with plastic wrap or a clean kitchen towel. Place in a warm place and allow dough to rise until doubled in size, about 1 hour.
Line a baking sheet with parchment paper and set aside. Turn dough onto a lightly floured surface and roll into a 13 x 9 rectangle. Cut evenly into squares and gently place in an even layer on the parchment lined baking sheet.
In a cast iron skillet, pour 2 to 3 cups of oil and heat over medium high.
When oil reaches about 350 degrees, add in the beignets, one by one, frying no more than 5 or 6 at a time.
Fry on each side until golden brown, about 1 ½ to 2 minutes per side.
Remove the beignets from the oil and place on a paper towel lined plate. While still hot, sprinkle with plenty of powdered sugar and enjoy hot and fresh!