Preheat oven to 375°F (190°C). Lightly grease a doughnut pan.
In a large bowl, whisk together flour, sugar, baking powder, cinnamon, nutmeg, and salt.
In a separate bowl, combine milk, eggs, vanilla extract, and melted butter. Mix well.
Add the wet ingredients to the dry ingredients and stir until just combined. Do not overmix.
Fill each doughnut mold about 3/4 full with the batter.
Bake in the preheated oven for 15-20 minutes, or until a toothpick inserted into the doughnuts comes out clean.
Let the doughnuts cool in the pan for 5 minutes, then transfer to a wire rack to cool completely. Dust with powdered sugar or glaze as desired.