In a small bowl, combine warm milk, yeast, and a teaspoon of sugar. Let it sit for 5 to 10 minutes until foamy.
In a large bowl, whisk together the remaining sugar, egg, melted butter, vanilla extract, and salt.
Add the yeast mixture and stir to combine.
Gradually add the flour, mixing until a soft dough forms. The dough should be slightly sticky but manageable.
Cover the bowl with a clean towel and let the dough rise in a warm place for about 1 hour, or until doubled in size.
Roll the dough out on a lightly floured surface to about 1/4-inch thickness.
Cut into 2 to 3-inch squares.
Heat oil in a deep pot to 350°F.
Fry the beignets in batches, turning once, until puffed and golden brown on both sides.
Remove with a slotted spoon and drain on paper towels.
Dust generously with powdered sugar and serve warm.