In a small bowl, dissolve 2 teaspoons of active dry yeast and 2 tablespoons of sugar into 1/2 cup of warm water (105-110°F). Allow it to sit for about 5-10 minutes until it becomes foamy.

In a large mixing bowl, whisk together the foamy yeast mixture, 1/2 cup of buttermilk, 1 large egg, and 2 tablespoons of melted unsalted butter until well combined. Gradually incorporate 2 cups of all-purpose flour, mixing until the dough forms a soft consistency.

Transfer the dough into a greased bowl and cover it with a clean kitchen towel or plastic wrap. Place it in a warm area to rise until doubled in size, approximately 1 hour.

Once the dough has risen, gently punch it down and roll it out on a lightly floured surface to about 1/2-inch thickness. Cut the dough into 2-inch squares.

In a deep pot or heavy pan, heat oil to 350°F for frying. Fry the dough squares in batches for 1-2 minutes per side or until they achieve a golden color.

Prepare the glaze by whisking together 1 cup of powdered sugar, 2 tablespoons of milk, and 1 teaspoon of vanilla extract until smooth. Dip each warm beignet into the glaze, allowing it to coat well.