01 - Combine warm water, 1 tablespoon of sugar, and yeast in a small bowl. Stir gently and let stand for 5 to 10 minutes until mixture becomes foamy and fragrant.
02 - Whisk together lukewarm milk, remaining granulated sugar, egg, softened butter, and vanilla extract in a large mixing bowl until fully incorporated. Pour in the activated yeast mixture and stir to combine.
03 - Gradually add flour and salt to the wet mixture, mixing continuously until a sticky, shaggy dough forms. Ensure all dry ingredients are fully absorbed.
04 - Turn dough onto a lightly floured work surface. Knead firmly for approximately 5 minutes, pushing dough away with heel of hand and folding back over itself, until smooth and elastic.
05 - Place dough in a greased bowl, turning once to coat all surfaces. Cover tightly with plastic wrap or a damp kitchen towel. Set in a warm, draft-free area for about 2 hours until doubled in bulk.
06 - Punch down risen dough to release air bubbles. Roll out on a lightly floured surface to 1/4-inch thickness. Using a sharp knife or pizza cutter, cut into 2 1/2-inch squares.
07 - Pour vegetable oil into a deep fryer or heavy-bottomed pot to a depth of at least 2 inches. Heat to 350°F, maintaining temperature throughout frying process.
08 - Fry beignet squares in small batches, turning frequently with a slotted spoon, until puffed and golden brown on all sides, about 2 to 3 minutes per batch. Avoid overcrowding the pot.
09 - Remove fried beignets with slotted spoon, allowing excess oil to drip back into pot. Transfer briefly to paper towel-lined surface to drain, then immediately coat generously with powdered sugar while still warm.