01 - In a large mixing bowl, whisk together flour, yeast, sugar, and salt until well blended.
02 - In a separate bowl, combine buttermilk, milk, eggs, and melted butter. Whisk until fully incorporated.
03 - Create a well in the center of the dry ingredients. Pour in the wet mixture and stir until a shaggy, sticky dough forms.
04 - Turn dough onto a lightly floured surface and knead for 6 to 8 minutes until smooth and elastic. Alternatively, use a stand mixer with dough hook for 5 minutes on medium speed.
05 - Place dough in a lightly oiled bowl, turning to coat all sides. 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 to release air. Roll out on a floured surface to approximately ½-inch thickness. Cut into 2-inch squares using a sharp knife or pastry cutter.
07 - Arrange squares on a parchment-lined baking sheet, leaving space between pieces. Cover loosely and let rest for 20 minutes.
08 - Heat vegetable oil in a deep pot or Dutch oven to 350°F. Working in batches, fry squares for 1 to 2 minutes per side until puffed and golden brown. Remove with a slotted spoon and drain on paper towels.
09 - While beignets fry, whisk powdered sugar, milk, and vanilla extract in a medium bowl until smooth and lump-free.
10 - Dip each warm beignet into the glaze, allowing excess to drip off. Place on a wire rack set over parchment paper to set. Serve immediately while warm.