Prepare the Yeast Mixture: Combine 1 cup of warm milk with 2 teaspoons of active dry yeast and a pinch of granulated sugar. Stir gently and let sit for 5-10 minutes until foamy.

Mix the Dough Ingredients: Whisk in 1/4 cup of granulated sugar, 1 large egg, 2 teaspoons of vanilla extract, 1/2 teaspoon of salt, and 2 tablespoons of melted butter until smooth.

Incorporate the Flour: Gradually add 3.5 to 4 cups of flour, mixing until a soft dough forms. Knead for 5-7 minutes on a floured surface.

Allow the Dough to Rise: Place kneaded dough in a greased bowl, cover with a towel, and let it rise for 1 to 1.5 hours until doubled in size.

Shape the Beignets: Punch down the dough, roll out to 1/4 inch thickness, and cut into 2 to 3-inch squares.

Heat the Oil: Heat vegetable oil in a heavy pot to 350°F (175°C), ensuring the right temperature for frying.

Fry the Beignets: Drop the squares of dough into hot oil, frying for about 1-2 minutes on each side until golden brown.

Drain and Dust: Remove beignets from oil, drain on paper towels, and dust generously with powdered sugar while warm.