01 - Dissolve yeast in warm water in a large bowl. Let stand for 5 minutes until foamy.
02 - Add sugar, salt, egg, and milk to the yeast mixture. Whisk until thoroughly combined.
03 - Stir in 2 cups flour, mixing until smooth. Add melted butter and remaining flour, stirring until soft dough forms.
04 - Turn dough onto lightly floured surface and knead for 4–5 minutes until smooth and elastic.
05 - Place dough in greased bowl, cover, and let rise in warm place for 1 hour until doubled in size.
06 - Punch down dough and transfer to floured surface. Roll into 1/2-inch thick rectangle. Cut into 1/2-inch wide by 3-inch long strips.
07 - Heat vegetable oil in deep skillet or pot to 350°F (175°C).
08 - Fry dough strips in batches, turning occasionally, until golden brown and puffed, approximately 2–3 minutes per batch.
09 - Remove with slotted spoon and drain on paper towels.
10 - Generously dust warm beignet fries with powdered sugar. Serve immediately.