1. Activate the Yeast: Warm the milk until it is warm but not hot or scalding, then dissolve the sugar in the milk. Sprinkle the yeast over it and let it sit for about 5 minutes until it becomes frothy.
2. Mix the Wet Ingredients: In a separate bowl, whisk together the eggs, melted butter, and vanilla extract. Combine this mixture with the activated yeast mixture until smooth.
3. Form the Dough: Gradually add the flour and salt to the wet mixture, stirring to form a soft dough. Knead the dough on a floured surface for 5-7 minutes until smooth and elastic, then place it in a greased bowl, cover, and let it rise in a warm place for 1-2 hours until doubled in size.
4. Shape and Rest: After the dough has risen, punch it down to release air, then roll it out to a half-inch thickness. Cut the dough into 2-inch squares and let the pieces rest for 10 minutes before frying.
5. Fry the Beignets: Heat oil in a deep pan to 360°F. Fry the dough squares in small batches, cooking for 1-2 minutes on each side until golden brown. Remove with a slotted spoon and drain on paper towels.
6. Dust and Serve: While the beignets are still warm, generously dust them with powdered sugar. Serve fresh and warm for the best experience, ideally with a cup of coffee or café au lait.