Activate the yeast: In a small bowl, dissolve 2 1/4 teaspoons of active dry yeast in 1/2 cup of warm water and 1 tablespoon of sugar (use from the 1/4 cup for the recipe). Let it sit for about 5-10 minutes until frothy.
Prepare the dough: In a large bowl, whisk together 1/2 cup of warm milk, 1/4 cup of melted unsalted butter, 2 large eggs, 1 teaspoon of vanilla extract, and the remaining 1/4 cup of sugar.
Combine wet and dry ingredients: Once the yeast mixture is frothy, add it to the milk mixture. Gradually add 4 cups of all-purpose flour and 1 teaspoon of salt until a soft dough forms.
Knead the dough: Turn the dough onto a floured surface and knead for about 5-7 minutes, or until smooth and elastic.
First rise: Place the dough in a greased bowl, cover it with a clean towel, and let it rise in a warm place for approximately 1-2 hours, or until doubled in size.
Roll out the dough: Once risen, punch down the dough and turn it out onto a lightly floured surface. Roll it out to about 1/4 inch thick.
Cut out the beignets: Using a sharp knife or pizza cutter, cut the dough into squares, about 2-3 inches wide.
Second rise: Place the cut-out squares on a floured baking sheet, cover, and let them rise for an additional 30 minutes.
Heat the oil: In a large, deep pot, heat enough vegetable oil for frying (at least 2 inches deep) to 350°F (175°C).
Fry the beignets: Fry the beignets in batches, about 2-3 minutes per side or until golden brown. Avoid overcrowding the pan to maintain oil temperature.
Drain and dust: Once fried, transfer beignets to a paper towel-lined plate to drain excess oil, then dust generously with powdered sugar before serving.