Start by gently warming your milk to a temperature between 105-115°F. In a small bowl, combine the warm milk, active dry yeast, and a pinch of sugar. Stir lightly and let the mixture sit for about 5-10 minutes until it becomes foamy.

In a mixing bowl, whisk together the foamy yeast mixture, remaining sugar, large egg, vanilla extract, salt, and melted unsalted butter until smooth.

Gradually add all-purpose flour to your wet mixture, stirring until a soft dough begins to form. Transfer the dough to a lightly floured surface and knead for about 5-7 minutes.

Place the kneaded dough in a greased bowl, cover with a damp kitchen towel or plastic wrap, and let it rise in a warm area for about 1 to 1.5 hours, or until doubled in size.

Once the dough has risen, punch it down gently to release the air and roll it out to approximately 1/4 inch thick. Cut the dough into 2-3 inch squares or rectangles.

In a heavy pot, heat vegetable oil to 350°F. Carefully drop a few squares of dough at a time into the hot oil, fry for 1-2 minutes on each side until golden brown.

Transfer the beignets onto a plate lined with paper towels to drain excess oil. Generously dust them with powdered sugar while they're still hot, and serve immediately.