Step 1: Mix the Dry Ingredients
In a large mixing bowl, whisk together flour, sugar, baking powder, and salt.

Step 2: Add Wet Ingredients
In a separate bowl, whisk the egg, milk, vanilla extract, and melted butter. Pour into the dry ingredients and stir until a soft dough forms. If the dough is too sticky, add a bit more flour one tablespoon at a time.

Step 3: Roll and Cut
Transfer the dough to a lightly floured surface. Roll it out to about ¼-inch thickness. Use a sharp knife or pastry cutter to slice the dough into squares or rectangles (about 2 inches each).

Step 4: Heat the Oil
In a heavy-bottomed pot or deep skillet, heat 2 inches of vegetable oil to 350°F (175°C). Use a thermometer to monitor the temperature for even frying.

Step 5: Fry the Beignets
Carefully drop a few beignet pieces at a time into the hot oil. Fry for about 1 minute per side or until golden brown and puffed. Do not overcrowd the pan.

Step 6: Drain and Dust
Use a slotted spoon to remove beignets and transfer to a paper towel-lined plate. Let cool slightly, then generously dust with powdered sugar.