Preheat the oven to 350°F (175°C). Spray or grease a doughnut pan or muffin tin really well (you can use muffin liners if you are using a muffin tin).
Mix together the flour, baking powder, cinnamon, nutmeg, and salt.
In another bowl, whisk together the egg, milk, melted butter (or oil), sugar and vanilla.
Stir the dry ingredients into the wet mixture until just combined.
Spoon the batter into the prepared donut pan, filling each one a nearly to the top. Bake for 15 to 17 minutes, until the donuts spring back to the touch or a toothpick comes out clean. Allow to cool for 5 minutes, then tap the doughnuts out onto a sheet pan. (If using a muffin tin, fill each well three-quarters full and bake for 20 to 25 minutes until they spring back to the touch or a toothpick comes out clean).