Combine the warm water and granulated sugar in a large bowl. Sprinkle the active dry yeast over the top. Let this mixture sit for 5 to 10 minutes until it becomes foamy. This shows the yeast is active.
Whisk the eggs, milk, melted butter, and vanilla extract into the yeast mixture until combined.
In a separate bowl, whisk together the all-purpose flour and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until a soft, sticky dough forms. You may not need all the flour.
Turn the dough out onto a lightly floured surface. Knead gently for about 5 minutes until the dough is smooth. It will remain slightly tacky.
Place the dough in a lightly oiled bowl, turning to coat. Cover the bowl with plastic wrap or a clean towel. Let the dough rise in a warm place for 1 to 1.5 hours, or until it has doubled in size.
Gently punch down the risen dough. On a lightly floured surface, roll the dough out to about 1/4 inch thickness.
Use a sharp knife or pizza cutter to cut the dough into 3-inch squares.
Pour vegetable oil into a deep, heavy-bottomed pot or deep fryer to a depth of about 2 inches. Heat the oil to 375°F (190°C). Use a thermometer to monitor the temperature closely.
Carefully drop 3 or 4 dough squares into the hot oil, ensuring you do not overcrowd the pot. Fry for 1 to 2 minutes per side, turning once, until they are puffed and golden brown.
Use a slotted spoon to remove the beignets from the oil. Place them on a wire rack lined with paper towels to drain excess oil.
While the beignets are still warm, generously dust them with powdered sugar. Serve immediately for the best texture.
Notes
For the best flavor, use real vanilla bean paste or extract in the dough.
Maintain the oil temperature between 365°F and 375°F. If the oil is too cool, the beignets will absorb too much oil and become greasy. If it is too hot, they will brown before cooking through.
You can use a simple glaze made from powdered sugar and milk instead of dry dusting if you prefer a wetter coating.