In a mixing bowl, whisk together flour, sugar, baking powder, salt, and nutmeg.
Add milk, eggs, melted butter, and vanilla extract; mix until smooth.
Lightly flour a clean surface and roll dough to 1/2-inch thickness.
Cut doughnuts using a doughnut cutter.
Heat oil to 350°F (175°C) and fry doughnuts 1 minute per side until golden brown.
Drain fried doughnuts on paper towels.
Alternatively, bake doughnuts at 375°F (190°C) for 10 to 12 minutes.
Prepare the glaze by mixing powdered sugar, cocoa powder, milk, melted butter, and vanilla until smooth.
Dip warm doughnuts into the chocolate glaze and allow to set before serving.
Enjoy warm with milk or coffee.