Proof the yeast: Dissolve yeast in warm water with a pinch of sugar. Wait 5 to 8 minutes until foamy.
Mix the wet ingredients: Whisk buttermilk, sugar, egg, melted butter, and salt together. Stir in the yeast mixture.
Form the dough: Add flour gradually and mix until a soft, slightly sticky dough forms.
Knead and rise: Knead for 5 minutes until smooth. Place in an oiled bowl, cover, and let rise for 1 hour until doubled.
Shape: Punch down dough, roll to 1/2 inch thickness, and cut into 2-inch squares.
Second rest: Place squares on parchment-lined tray, cover loosely, and rest for 20 to 30 minutes.
Fry: Heat oil to 360°F. Fry in small batches, 1 to 2 minutes per side, until deep golden. Drain on a wire rack.
Glaze: Whisk powdered sugar, milk, and vanilla until smooth. Drizzle over warm beignets and let set for 10 minutes.