In a large bowl, whisk together flour, sugar, baking powder, and salt.
In a separate bowl, combine milk, egg, vanilla extract, and melted butter.
Pour the wet ingredients into the dry ingredients and mix until a dough forms.
Turn the dough onto a floured surface and knead gently for about 2 minutes.
Roll the dough out to about 1/4 inch thick.
Cut into squares or rectangles, about 2-3 inches in size.
Heat oil in a deep fryer or heavy pot to 350°F (175°C).
Fry the beignets in batches, about 2-3 minutes on each side until golden brown.
Remove and drain on paper towels.
Dust generously with powdered sugar before serving.