Preheat oven to 350 degrees. Spray 1 doughnut pan with nonstick spray.
In a large bowl whisk flour, sugar, baking powder, cinnamon, nutmeg and salt.
In a separate bowl whisk egg, milk, melted butter and vanilla. Add the wet ingredients into the dry and stir until just combined.
Spoon the batter into the doughnut pan*, filling each a little more than three-quarters full. Bake for approximately 15 minutes, until a toothpick inserted into the doughnuts comes out clean. Allow to cool for 5 minutes and then remove the doughnuts from the pan.
For the topping:
Melt 4 tablespoons of butter in a sauté pan.
Combine the sugar and cinnamon in a small bowl.
Dip each doughnut in the melted butter and then into the cinnamon sugar.Repeat on both sides.
Notes
*A note on using up extra batter: After filling my doughnut pan, which holds 6 doughnuts, I still had a little extra batter leftover. I put the remaining batter in my mini muffin pan and I was able to make about 6 "doughnut holes." These bake in a little less time than the doughnuts (about 8-10 minutes). Dip them in the melted butter and roll in the cinnamon sugar topping also.