01 - Combine warm water, milk, and yeast in a small bowl. Let stand for 5 minutes until foamy and activated.
02 - Whisk flour, sugar, and salt together in a large mixing bowl until thoroughly combined.
03 - Add eggs, melted butter, vanilla extract, and yeast mixture to dry ingredients. Mix until a sticky, cohesive dough forms.
04 - Turn dough onto lightly floured surface and knead for 5-7 minutes until smooth and elastic.
05 - Place dough in greased bowl, cover with plastic wrap or damp towel. Let rise in warm place until doubled, approximately 1-1.5 hours.
06 - Punch down risen dough. Roll out to 1/4-inch thickness on floured surface. Cut into 2-inch squares.
07 - Heat oil in heavy-bottomed pot or deep fryer to 350°F.
08 - Fry beignets in batches of 4-6 for 1-2 minutes per side until puffed and golden brown.
09 - Remove with slotted spoon and drain on paper towels. Generously dust with powdered sugar while warm. Serve immediately.