Preheat oven to 350 degrees Fahrenheit.
In a large bowl, mix together dry ingredients: Flour, baking powder, salt, nutmeg. Add wet ingredients: Butter, sugar, egg, milk, vanilla extract. Mix together thoroughly.
Grease baking pan or muffin tin. Fill each cup 2/3 full.
Bake at 350 degrees F for 15 to 20 minutes, or until doughnuts are golden. While the doughnuts are baking, prepare the coating: melt 3 tablespoons butter, mix in 1/4 cup sugar and 1/2 teaspoon cinnamon.
Roll doughnuts in coating, and serve.