Mix flour, sugar, instant yeast, and salt in a large bowl. Stir in lukewarm milk, egg, cooled melted butter, and vanilla until a shaggy dough forms. Rest covered for 5 minutes, then knead 8–10 minutes by hand, or about 6–8 minutes with a dough hook on low, until smooth, elastic, and slightly tacky. Dust lightly only as needed.

Return to the bowl, cover, and let rise in a comfortably warm spot until roughly doubled, about 60–90 minutes. Use the dough's expansion rather than the clock alone.

Gently deflate the dough and divide into 12 equal pieces, about 64 g each. Round each piece into a ball, cover, and rest 10 minutes. Poke a hole through each center and gently stretch into a 3 1/2–4 inch ring with a hole about 1 1/2 inches wide. Set on lightly floured parchment, leaving space between rings.

Cover loosely and let rise 30–45 minutes, until noticeably puffy and a gentle indentation springs back slowly. Do not rush this rise. Keep the rings covered until ready to fry.

Set a wire rack over a rimmed tray. Add about 2 inches of canola oil to a deep, heavy pot, keeping it no more than half full. Clip on a frying thermometer and heat to 350°F. Keep a metal lid nearby and never leave heating oil unattended.

Lift 2–3 rings carefully with a broad floured spatula and lower gently into the oil. Fry about 1–2 minutes per side, until evenly golden. Maintain 350–360°F and allow the oil to recover between batches. Lift with a slotted spoon onto the rack. Cool the first donut enough to split it and check that its center is fully set with no wet dough; adjust subsequent frying time as needed.

Whisk powdered sugar, 60 g milk, and 4 g vanilla until smooth. Once the donuts are warm rather than piping hot, dip both sides in glaze, let excess drip off, and return to the rack. Let the coating set for about 10–15 minutes before serving.