Step 1: Activate the Yeast

1. In a small bowl, combine the warm water, 1 tablespoon of sugar, and the active dry yeast. Stir gently and let the mixture sit for 5-10 minutes until it becomes foamy. This indicates that the yeast is activated.

Step 2: Prepare the Dough

1. In a large mixing bowl, whisk together the warm milk, melted butter, ¼ cup granulated sugar, egg, vanilla extract, and salt. Stir in the activated yeast mixture.
2. Gradually add the flour, one cup at a time, mixing with a wooden spoon or a stand mixer fitted with a dough hook. Once the dough begins to come together, knead it by hand or with the dough hook for 5-7 minutes, adding more flour as needed until the dough is smooth and slightly tacky but not sticky.

Step 3: Let the Dough Rise

1. Form the dough into a ball and place it in a greased bowl. Cover with plastic wrap or a clean kitchen towel and let it rise in a warm, draft-free area for about 1 ½ to 2 hours, or until it has doubled in size.

Step 4: Roll and Cut the Dough

1. Once the dough has risen, punch it down to release any air bubbles. Transfer it to a lightly floured surface and roll it out to about ¼-inch thickness.
2. Use a sharp knife or a pizza cutter to cut the dough into 2 ½ to 3-inch squares. You can also use a round cookie cutter if you prefer circular beignets.

Step 5: Heat the Oil

1. In a large, heavy-bottomed pot or Dutch oven, heat the vegetable oil to 350°F (175°C). Use a kitchen thermometer to monitor the temperature and ensure consistent frying.

Step 6: Fry the Beignets

1. Carefully add 3-4 pieces of dough to the hot oil, being careful not to overcrowd the pot. Fry for about 1-2 minutes on each side, or until the beignets are puffed and golden brown.
2. Use a slotted spoon to transfer the fried beignets to a paper towel-lined plate to drain excess oil. Repeat with the remaining dough.

Step 7: Dust with Powdered Sugar

1. Generously dust the warm beignets with powdered sugar on both sides. For best results, use a fine-mesh sieve or sifter for an even coating.
2. Serve immediately while warm and enjoy with a hot cup of coffee, tea, or hot chocolate.