Mix the flour, sugar, yeast and salt in a large bowl. Add the lukewarm milk, egg, softened butter and vanilla. Stir into a soft dough, then knead for 8–10 minutes until smooth and elastic. The dough should feel slightly tacky; add dusting flour sparingly.

Cover the bowl and let the dough rise until approximately doubled, about 60–90 minutes depending on room temperature. Lightly flour the worktop, gently press out the gas and roll to about 1/2 inch thick.

Cut rings with a 3-inch cutter and a small center cutter. Put them on parchment squares, cover loosely and let rise for 30–45 minutes until visibly puffy. Reroll scraps once, let them rest briefly and cut additional rings. Expect about 12 rings plus holes; yield varies with cutter size.

Heat 2 inches of oil in a deep, heavy pot to 350°F (175°C), checking with a frying thermometer. Keep the pot no more than half full. Lower in 2–3 rings carefully without crowding; remove any parchment with tongs as it releases. Fry about 60–90 seconds per side, until evenly golden and cooked through. Fry holes separately because they finish faster.

Lift donuts onto a rack over a tray. Let the oil recover to 350°F (175°C) between batches; adjust the heat as needed. Break open the first cooled donut to check that the center is cooked, not gummy.

Stir the powdered sugar, 3 tablespoons milk and vanilla into a smooth glaze. Add the extra milk a teaspoon at a time only if needed for a thin coating. Dip the warm, not piping-hot, donuts and return to the rack. Let the glaze set for 10–15 minutes before serving.