In a large bowl, combine warm milk, yeast, and a pinch of sugar. Let stand for 5-10 minutes until foamy.
Whisk in the remaining granulated sugar, egg, vanilla extract, salt, and melted butter into the yeast mixture.
Gradually add the flour, mixing until a soft dough forms. Knead on a lightly floured surface for 5-7 minutes until smooth and elastic.
Place the dough in a greased bowl, cover, and let rise in a warm place for 1 to 1.5 hours, or until doubled in size.
Punch down the dough, then roll it out on a lightly floured surface to about 1/4 inch thickness. Cut into 2-3 inch squares or rectangles.
Heat vegetable oil in a deep fryer or large pot to 350°F (175°C). Fry beignets a few at a time for 1-2 minutes per side, until golden brown and puffed.
Remove beignets with a slotted spoon and drain on paper towels. Dust generously with powdered sugar before serving warm.
