Mix the flour, instant yeast, granulated sugar, salt and nutmeg. Add the lukewarm milk and egg, mix to a rough dough, then knead in the softened butter.

Knead for about 8 minutes until smooth, elastic and slightly tacky. Cover the bowl and let rise in a comfortably warm place for about 60–90 minutes, until doubled; use the dough's expansion as the main cue.

Roll on a lightly floured counter to about 1/2-inch thickness. Cut approximately eight 3-inch rings with 1-inch holes, gently rerolling scraps once. Put on lightly floured parchment and cover loosely. Let rise about 30–45 minutes until distinctly puffy.

Set a rack over a tray. Heat 2 inches of oil in a deep heavy pot to 350°F (175°C), checking with a deep-fry thermometer. Keep ample empty space above the oil and never leave it unattended.

Carefully lower two or three rings at a time into the oil. Fry about 60–90 seconds per side until evenly golden. Open the first cooled test ring to check that the center is cooked, with no sticky raw dough. Let the oil return near 350°F between batches.

Fry the holes separately, turning as needed, for about 1 minute total until cooked through and golden. Lift the donuts onto the rack to drain.

Whisk the powdered sugar, milk and vanilla until smooth and pourable. Dip the warm, not piping-hot, rings and holes lightly, let excess drip off and return to the rack. Allow the glaze to set briefly before serving.