In a small bowl, stir together the warm water, 1 teaspoon of the granulated sugar (taken from the 1/4 cup), and the active dry yeast. Let it stand for 5–10 minutes, until the mixture is foamy and creamy on top. If it does not foam, discard and start again with fresh yeast and correctly warmed liquids.

In a large mixing bowl or the bowl of a stand mixer, whisk together the warm milk, remaining granulated sugar, eggs, melted and slightly cooled butter, vanilla, and salt until well combined. The mixture should be pale yellow and the sugar mostly dissolved.

Add the foamy yeast mixture to the bowl with the wet ingredients and stir to combine. Add 2 cups of the flour and mix with a wooden spoon or the paddle attachment on low speed until smooth. Gradually add the remaining 1 1/2 cups flour, about 1/2 cup at a time, mixing until a soft dough forms that is slightly sticky but pulls away from the sides of the bowl. If extremely sticky, add 1–2 tablespoons more flour, but avoid adding too much to keep the dough tender.

Turn the dough out onto a lightly floured surface or switch to a dough hook on your stand mixer. Knead by hand for 6–8 minutes, or with the mixer on medium-low for 5–6 minutes, until the dough is smooth, stretchy, and elastic. It should feel soft and a bit tacky but not gluey, and should slowly bounce back when pressed.

Lightly oil a clean bowl and place the dough inside, turning once to coat it in a thin layer of oil. Cover the bowl with plastic wrap or a clean kitchen towel and let the dough rise in a warm, draft-free spot for 1 to 1 1/2 hours, or until about doubled in size. If your kitchen is cool, you can place the bowl in a turned-off oven with the light on.

Gently punch down the risen dough to release excess gas. Turn it out onto a lightly floured surface and roll it into a rectangle about 1/4 inch thick. Using a pizza cutter or sharp knife, cut the dough into 2–2 1/2 inch squares. They do not need to be perfectly even.

Transfer the dough squares to a lightly floured baking sheet or parchment-lined tray, leaving a little space between them. Cover loosely with a kitchen towel and let rest for 20–30 minutes, until slightly puffy.

In a heavy-bottomed pot or Dutch oven, pour in 2–3 inches of vegetable or canola oil. Heat the oil to 360–370°F, using a deep-fry thermometer if possible. Without a thermometer, test by dropping in a small scrap of dough: it should sizzle immediately, float in about 5 seconds, and turn golden in about 45–60 seconds. If it browns in about 10 seconds, the oil is too hot.

Working in batches, carefully place 3–5 dough squares into the hot oil at a time, avoiding overcrowding. Fry for 1–2 minutes per side, turning once, until puffed and deep golden brown. Use a slotted spoon to transfer the fried beignets to a paper towel–lined plate or cooling rack to drain. If they are dark but flat, reduce the oil temperature slightly.

While the beignets are still warm, place powdered sugar in a fine-mesh sieve (optionally mixed with a pinch of cinnamon) and generously dust the tops and sides until well coated. Serve immediately with extra powdered sugar on the side. Beignets are best within 15–20 minutes of frying.