Mix warm milk (about 110°F/43°C) and yeast with a pinch of sugar; let foam
Whisk flour, sugar, salt, and nutmeg (optional)
Add eggs, melted butter, and yeast mixture
Knead until smooth and elastic
Cover and proof until doubled in size
Punch down and roll dough to about 1/2-inch thickness
Cut donuts and donut holes
Cover and proof until puffy
Heat oil to about 350°F/175°C
Fry donuts until golden on both sides
Remove and drain on a wire rack or paper towels
Cool slightly before glazing or coating
Glaze: whisk powdered sugar with milk or water and a bit of vanilla; dip tops
Alternative coatings: cinnamon sugar, chocolate glaze, or powdered sugar