Preheat the oven to 350 degrees and spray 2 doughnut pans well.
Sift the flour, sugar, baking powder, cinnamon, nutmeg, and salt. In a separate bowl, whisk together the egg, milk, melted butter, and vanilla. Stir the wet mixture into the dry ingredients until just combined.
Spoon the batter into the baking pans, filling each approximately three-quarters full. Bake for 17 minutes or until a toothpick comes out clean. Cool for 5 minutes, then remove.