01 - Whisk together flour, yeast, sugar, and salt in a large mixing bowl until thoroughly combined.
02 - In a separate bowl, blend buttermilk, milk, eggs, and melted butter until uniform.
03 - Create a well in the dry ingredients. Pour in the wet mixture and stir until a shaggy, sticky dough forms.
04 - Turn dough onto a floured surface and knead for 6 to 8 minutes until smooth and elastic. Alternatively, use a stand mixer with dough hook for 5 minutes.
05 - Place dough in a lightly oiled bowl, cover with plastic wrap or a damp towel, and let rise in a warm, draft-free area for 60 to 75 minutes until doubled in size.
06 - Punch down risen dough and roll out on a floured surface to ½-inch thickness. Cut into 2-inch squares using a sharp knife or pastry cutter.
07 - Arrange squares on a parchment-lined baking sheet, cover loosely with a towel, and let rest for 20 minutes to puff slightly.
08 - Heat oil in a deep pot or Dutch oven to 350°F. Fry beignets in batches for 1 to 2 minutes per side until puffed and golden brown. Remove with a slotted spoon and drain on paper towels.
09 - Whisk powdered sugar, milk, and vanilla extract in a bowl until completely smooth and free of lumps.
10 - While beignets are still warm, dip each square into the glaze, allowing excess to drip off. Transfer to a wire rack set over parchment paper until glaze sets.