In a large bowl, whisk together the flour, sugar, salt, and dry yeast.
Add the warm milk, beaten eggs, and vanilla extract, then begin to combine the dough.
Melt the butter and gradually add it to the dough, kneading continuously until you have a smooth, elastic dough.
Cover the dough with a kitchen towel and let it rise in a warm place for about 1 hour, or until doubled in size.
On a lightly floured surface, roll out the dough to about 1 cm thickness, then cut into squares or rectangles.
In a deep pot, heat the oil to 350°F (180°C). Check the oil temperature with a small piece of dough – if it immediately rises to the surface and gently fries, the oil is ready.
Fry the beignets for 2-3 minutes on each side, until golden brown. Do not overcrowd the pot, so the oil temperature doesn't drop.
Remove the fried beignets to a paper towel-lined plate to absorb excess oil. Let them cool slightly.
Sprinkle generously with powdered sugar and serve warm.
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