Preheat your oven to 350°F.
In a large mixing bowl, whisk together the flour, cinnamon, nutmeg, granulated and brown sugar, baking powder, and salt.
In a separate bowl, whisk together the milk, eggs, melted butter, and vanilla extract until well combined.
Pour the wet ingredients into the dry ingredients and stir until just combined. Be careful not to overmix.
Lightly grease a donut pan with nonstick cooking spray. Spoon the batter into a piping bag or a resealable plastic bag with the corner snipped off. Pipe the batter into the prepared donut pan, filling each cavity about 2/3 full.
Bake the donuts for 13 minutes, or until they are lightly golden and a toothpick inserted into the center comes out clean
For the glaze, whisk together the powdered sugar, milk, and vanilla extract in a small bowl until smooth and well combined. Adjust the consistency with more milk if needed.
Transfer the baking tin to a wire rack and allow the donuts to cool completely (about 1 hour).
Dip the cooled donuts into the glaze, allowing any excess to drip off. Let the glaze set for a few minutes before serving.