01 - Whisk together flour, yeast, sugar, and salt in a large mixing bowl until thoroughly blended.
02 - In a separate bowl, whisk buttermilk, milk, eggs, and melted butter until fully incorporated.
03 - Create a well in the center of dry ingredients. Pour wet mixture into the well and stir until shaggy dough forms.
04 - Turn dough onto floured surface and knead for 6 to 8 minutes until smooth and elastic, or use stand mixer with dough hook for 5 minutes.
05 - Place dough in lightly oiled bowl, cover, and let rise in warm area for 60 to 75 minutes until doubled in size.
06 - Punch down risen dough. Roll out on floured surface to ½-inch thickness. Cut into 2-inch squares using sharp knife or pastry cutter.
07 - Arrange squares on parchment-lined tray, cover loosely, and let rest for 20 minutes.
08 - Heat oil in deep pot to 350°F. Fry beignets in batches for 1 to 2 minutes per side until puffed and golden brown. Remove with slotted spoon and drain on paper towels.
09 - Whisk powdered sugar, milk, and vanilla extract in bowl until smooth and no lumps remain.
10 - While beignets are still warm, dip each square into glaze, allowing excess to drip off. Place on cooling rack to set for 5 minutes before serving.