Whisk together flour, sugar, baking powder, salt, and nutmeg in a bowl.
Add milk, eggs, melted butter, and vanilla extract; mix until smooth.
Lightly flour a surface and roll out the dough to ½-inch thickness.
Cut dough using a doughnut cutter or round cutter.
Heat oil to 350°F (175°C) and fry doughnuts for 1 minute per side until golden brown.
Drain fried doughnuts on paper towels.
Alternatively, bake doughnuts at 375°F (190°C) for 10–12 minutes.
Mix powdered sugar, cocoa powder, milk, melted butter, and vanilla until smooth to make the glaze.
Dip warm doughnuts into chocolate glaze and let set.
Serve warm with milk or coffee.