In a large bowl, combine the flour, granulated sugar, active dry yeast, and salt. Mix well to distribute the yeast and sugar evenly through the flour.

In a separate bowl, whisk together the warm milk (about 110°F), melted unsalted butter, and pure vanilla extract. Ensure the mixture is warm to the touch, not hot, to activate the yeast without killing it.

Gradually pour the wet ingredients into the dry ingredients while stirring with a wooden spoon until a dough begins to form. Continue to mix until all ingredients are thoroughly combined.

Transfer the dough onto a lightly floured surface and knead for about 8-10 minutes until the dough is smooth and elastic. If the dough is too sticky, sprinkle a little more flour as needed.

Place the kneaded dough into a greased bowl, turning it once to coat all sides with oil. Cover the bowl with a clean cloth and let it rise in a warm, draft-free area for about 1.5 hours, or until it has doubled in size.

Once risen, punch down the dough and roll it out on a floured surface to about ¼ inch thickness. Cut the dough into 2×2 inch squares using a sharp knife or pizza cutter.

Pour about 2 inches of vegetable oil into a deep fryer or large pot. Heat the oil to 360°F, using a candy or deep-fry thermometer to ensure accuracy.

Fry the beignets in batches, careful not to overcrowd the pot, for about 1 to 2 minutes on each side, or until they are puffed up and golden brown. Use a slotted spoon to remove them from the oil and drain on paper towels.

Dust the freshly fried beignets with powdered sugar and serve immediately.

Allow the beignets to cool slightly on a wire rack if not served immediately.