Proof yeast: In a small bowl, whisk warm milk with 1 tablespoon of the sugar and the active dry yeast. Let stand 5–10 minutes until foamy. If there is no foam, discard and start with fresh yeast.
Mix wet base: In a large bowl, whisk the egg, remaining sugar, melted butter, and vanilla until smooth and fragrant.
Build dough: Stir the foamy yeast mixture into the bowl. Add half the flour and the salt; mix to a thick batter. Add remaining flour gradually until a soft, slightly sticky dough forms.
Knead: Turn onto a lightly floured surface and knead 4–6 minutes (or mix with a dough hook about 5 minutes) until smooth and elastic.
First rise: Place dough in a lightly greased bowl, cover, and let rise in a warm spot 60–90 minutes, or until doubled in size.
Roll and cut: Gently deflate, then roll to 1/4–1/2 inch thickness. Cut into 2-inch squares (or circles). Rest scraps 5 minutes and recut.
Heat oil: Pour 2–3 inches of vegetable oil into a deep pot and heat to 350°F (175°C). Keep the thermometer clipped to maintain temperature.
Fry: Fry beignets in small batches 1–2 minutes per side until deep golden and puffed. Avoid crowding to keep oil hot.
Drain: Transfer with a slotted spoon to a paper towel–lined rack to drain briefly.
Finish: While warm, dust generously with powdered sugar (and a pinch of vanilla bean powder or cinnamon if using). Serve immediately with coffee, hot chocolate, honey, chocolate sauce, or fruit preserves.
Optional filling: Once slightly cooled, poke a small hole and pipe in jam, pastry cream, or chocolate for stuffed beignets.
Storage: Keep at room temperature up to 2 days; rewarm at 350°F for 5–6 minutes. Freeze up to 2 months and reheat from frozen until hot and crisp, then re-dust with sugar.