In a large bowl, or the bowl of a stand mixer, combine the warm milk and yeast. Let stand for 5-10 minutes, or until foamy.

Add the sugar, salt, egg, vanilla extract, and melted butter to the yeast mixture. Stir well to combine.

Gradually add the flour, one cup at a time, mixing until a shaggy dough forms.

If using a stand mixer, knead the dough with the dough hook attachment for 5-7 minutes, or until smooth and elastic. If kneading by hand, turn the dough out onto a lightly floured surface and knead for 8-10 minutes, or until smooth and elastic. The dough should be slightly sticky but manageable.

Place the dough in a lightly oiled bowl, turning to coat. Cover with plastic wrap and let rise in a warm place for 1-1 1/2 hours, or until doubled in size.

Once the dough has doubled, gently punch it down to release the air.

Turn the dough out onto a lightly floured surface and roll it out to about 1/2 inch thickness.

Using a pizza cutter or sharp knife, cut the dough into 2-3 inch squares.

Place the cut beignets on a baking sheet lined with parchment paper. Cover loosely with plastic wrap and let rise for another 30-45 minutes.

While the beignets are rising, heat the vegetable oil in a large, heavy-bottomed pot or deep fryer to 350-375°F (175-190°C).

Carefully drop the beignets, a few at a time, into the hot oil. Fry for 2-3 minutes per side, or until golden brown and puffed up.

Remove the beignets with a slotted spoon and place them on a wire rack lined with paper towels to drain excess oil.

While the beignets are still warm, generously dust them with powdered sugar.