Preheat the oven to 350 degrees.Spray 2 doughnut pans with oil.Whisk (or sift) the flour, sugar, cinnamon, nutmeg, baking powder and salt.In another bowl, whisk together the egg, milk, melted butter, and vanilla.Stir the wet ingredients into the dry ingredients until just combined.Spoon the batter into the baking pans, filling each one a little more than ¾ full.Bake for 17 minutes, until a toothpick comes out clean.Cool on a rack for 5 minutes, then remove the doughnuts.Melt the butter either on the stovetop or microwave.In a small bowl, combine the sugar and cinnamon.Dip each doughnut in the butter and sit on a rack over a sheet pan. When the donuts are all buttered, roll in the cinnamon sugar so they're coated all over. I found that dipping in butter and immediately in sugar left globs of cinnamon sugar. when I let the butter soak in for a few minutes they sugared much better.