01 - Dissolve the yeast in warm water in a large bowl. Let stand for 5 minutes until foamy.
02 - Add the sugar, salt, egg, and milk to the yeast mixture. Whisk until thoroughly combined.
03 - Stir in 2 cups of flour, mixing until smooth. Add the melted butter and remaining flour, stirring until a soft dough forms.
04 - Turn the dough onto a lightly floured surface and knead for 4–5 minutes until smooth and elastic.
05 - Place the dough in a greased bowl, cover, and let rise in a warm place for 1 hour until doubled in size.
06 - Punch down the dough and transfer to a floured surface. Roll into a rectangle about 1/2 inch thick. Cut into strips about 1/2 inch wide and 3 inches long to form fries.
07 - Heat vegetable oil in a deep skillet or pot to 350°F.
08 - Fry the dough strips in batches, turning occasionally, until golden brown and puffed, about 2–3 minutes per batch.
09 - Remove with a slotted spoon and drain on paper towels. While still warm, generously dust the beignet fries with powdered sugar. Serve immediately.