In a large mixing bowl, whisk together the flour, sugar, baking powder, and salt until well combined.
In another bowl, beat the eggs and then add the milk and vanilla extract. Mix well.
Pour the wet ingredients into the dry ingredients and stir until just combined. The dough will be a bit sticky.
Lightly flour a clean surface and transfer the dough onto it. Gently knead the dough for about 1 minute until it becomes smoother.
Roll out the dough to about 1/4 inch thickness. Use a sharp knife or pizza cutter to cut it into squares (approximately 2x2 inches).
Heat vegetable oil in a deep pot or fryer to 350°F (175°C). Carefully drop a few squares of dough into the hot oil, making sure not to overcrowd the pot.
Fry the beignets for approximately 2-3 minutes on each side, or until they are golden brown. Use a slotted spoon to transfer them to a paper towel-lined plate to drain excess oil.
While still warm, dust the beignets generously with powdered sugar before serving.
Notes
Serve the beignets in a stacked tower on a decorative plate, with a small bowl of extra powdered sugar on the side for dipping.