Warm the milk and wake the yeast: Heat milk until just warm to the touch, about 105–110°F (40–43°C). Stir in a pinch of sugar and the yeast. Let it sit 5–10 minutes until foamy. If it doesn’t foam, start again with fresh yeast.

Mix the dough: In a large bowl, whisk flour, sugar, and salt. Add the foamy yeast mixture, eggs, vanilla, and softened butter. Mix with a dough hook or wooden spoon until a shaggy dough forms.

Knead to smooth: Knead 6–8 minutes (stand mixer on medium-low, or by hand on a lightly floured surface) until the dough is soft, elastic, and slightly tacky but not sticky. Add a sprinkle of flour only if it clings heavily to your hands.

First rise: Place the dough in a lightly greased bowl, cover, and let rise in a warm spot until doubled, about 1–1½ hours. A slow overnight rise in the fridge also works and adds flavor.

Roll and cut: Turn the dough onto a floured surface. Roll to about ½ inch (1.25 cm) thick. Cut out large rounds with a 3-inch cutter, then punch out centers with a 1-inch cutter. Gently gather scraps, rest 5 minutes, and re-roll once.

Second rise: Place rings on parchment-lined sheets lightly dusted with flour. Cover loosely and let rise until puffy, 30–45 minutes. They should look airy and feel delicate if you lift a corner of parchment.

Heat the oil: In a heavy pot, heat 2–3 inches of oil to 350–360°F (175–182°C). Keep a thermometer clipped on and adjust the heat often. Too cool and they’ll get greasy; too hot and they’ll brown before cooking through.

Fry the doughnuts: Working in batches, slide doughnuts into the oil by lifting on parchment squares, then peeling the paper away with tongs. Fry 45–60 seconds per side until golden. Drain on a rack set over a sheet pan. Fry the holes too—they cook faster.

Make the glaze: Whisk powdered sugar, a splash of warm milk, vanilla, a pinch of salt, and a teaspoon of corn syrup (optional) until smooth and pourable. Aim for a thin ribbon that disappears back into the bowl in 2–3 seconds.

Dip while warm: When doughnuts are warm but not hot, dunk each one fully in glaze, flip to coat, then let excess drip back. Set on the rack to set for 10–15 minutes until glossy.

Serve fresh: Enjoy within a few hours for peak texture, or see storage tips below.