Make the dough: combine warm milk, yeast, sugar, and let it foam
Mix in eggs, melted butter, flour, salt, and nutmeg (optional)
Knead until smooth and elastic
Cover and let rise until doubled in size
Roll out and cut donuts using a donut cutter
Cut donut holes and place all pieces on a parchment-lined tray
Cover and let rise again until puffy
Heat oil to about 350°F / 175°C
Fry donuts until golden brown on both sides
Remove and drain on a wire rack
Coat with glaze or sugar while warm
Glaze: whisk powdered sugar with milk or water and add vanilla (optional)
Optional toppings: sprinkles, cinnamon sugar, or chocolate coating