Preheat the oven to 350°F. Spray 2 doughnut pans well.
Into a large bowl, sift together the flour, sugar, baking powder, cinnamon, nutmeg, and salt. In a small bowl, whisk together the egg, milk, melted butter, and vanilla. Stir the wet mixture into the dry ingredients until just combined.
Bake for 17 minutes pr until toothpick comes out clean.
For the topping, melt the 8 tablespoons of butter in an 8-inch saute pan. Combine the sugar and cinnamon in a small bowl. Dip each doughnut first in the butter and then in the cinnamon sugar, either on one side or both sides.