In a medium bowl, combine warm milk, active dry yeast, and a pinch of granulated sugar. Let stand for 5 to 10 minutes until foamy.

Whisk in granulated sugar, egg, vanilla extract, salt, and melted unsalted butter until blended.

Gradually add all-purpose flour until a soft, slightly sticky dough forms. Knead for 5 to 7 minutes until smooth and elastic.

Place dough in a greased bowl, cover with kitchen towel, and let rise for 1 to 1.5 hours until doubled in size.

Punch dough down, roll out to about 1/4 inch thick, and cut into squares or rectangles, approximately 2 to 3 inches wide.

In a deep pot, heat vegetable oil to 350°F (175°C). Fry beignets in small batches for 1 to 2 minutes on each side until golden brown.

Remove beignets with a slotted spoon, drain on paper towels, and dust with powdered sugar while warm.