In a large mixing bowl, stir warm milk with sugar and sprinkle yeast on top. Let it sit until foamy, about 10 minutes.
Whisk in the eggs, melted butter, and vanilla.
Gradually add flour and salt, mixing until a soft dough forms.
Knead the dough on a floured surface for about 8 minutes. It should feel smooth but springy.
Place the dough in a greased bowl, cover it, and let it rise in a warm spot for 1 to 2 hours, until doubled.
Punch down the dough, then roll it out to 1/2-inch thickness and cut into squares or circles.
Heat oil to 350°F in a deep pan. Fry the beignets in batches until golden brown, about 1-2 minutes per side.
Drain them on paper towels and give them a generous snowing of powdered sugar.
Serve warm and try not to eat them all in one sitting.