Bloom the yeast: In a small bowl combine warm water (¼ cup) with 1 tsp sugar. Sprinkle yeast over the surface and let sit 5–8 minutes until foamy. If no foam, discard and use fresh yeast.
Combine dry ingredients: In a large bowl whisk 3½ cups (437 g) of flour with the remaining granulated sugar and salt.
Mix wet ingredients: Whisk warm milk, melted butter, egg, and vanilla in a separate bowl. Add the foamy yeast mixture to the milk mixture and stir.
Form the dough: Pour wet into dry and mix until a shaggy dough forms. If using a stand mixer, mix on low with a dough hook until combined.
Knead: Add remaining flour (up to ½ cup) a tablespoon at a time only if the dough is excessively sticky. Knead by machine 6–8 minutes or by hand 8–10 minutes until smooth and elastic. Dough should be soft and slightly tacky.
First rise: Shape dough into a ball, place in a lightly oiled bowl, cover, and let rise in a warm spot until doubled (about 60–90 minutes).
Roll & cut: Gently deflate dough, transfer to a floured surface, and roll to about ½ inch (1.2 cm) thickness. Cut into 3×3-inch (7.5–8 cm) squares (or preferred size). Place pieces on a floured tray, cover loosely, and rest 20–30 minutes to puff slightly.
Heat the oil: In a heavy pot or deep fryer heat oil to 350–360°F (175–182°C). Use a thermometer to maintain steady temperature. Test with a small dough scrap — it should float and puff immediately.
Fry: Fry 3–4 beignets at a time (don’t overcrowd) about 1½–2 minutes per side until golden and puffed, flipping once. Total per piece ~3–4 minutes. Keep oil steady at 350–360°F.
Drain & sugar: Transfer to a wire rack set over a sheet pan to drain and cool slightly. Immediately sift powdered sugar over the hot beignets; be generous.
Serve: Serve warm with extra powdered sugar and optional dipping sauces (salted caramel, chocolate ganache, jam).