Lightly grease 2 (6-cavity) doughnut pans with cooking spray and set aside.You can also use a 12-cup muffin tin, if you don't have a donut pan.
In a mixing bowl, combine flour, baking powder, salt, cinnamon, and nutmeg; whisk to combine and set aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream the butter, brown sugar, and granulated sugar on medium speed until light and fluffy, about 3 to 4 minutes.
Add the eggs, one at a time, and mix until well incorporated after each addition, scraping the bowl as necessary. Then, add in the vanilla.
Set mixer to low speed and slowly add in the flour mixture; continue to mix until incorporated.
With the mixer running, add the apple cider in a slow, steady stream and continue to mix until combined, scraping down the bowl as needed.
Spoon the batter into previously prepared doughnut pans, filling them about ⅔ of the way up. You can also spoon the batter into a piping bag and pipe the batter into the molds.
Bake in the oven for about 12 to 15 minutes, or until golden brown and a toothpick inserted into the center of the thickest portion of the doughnut comes out with a few crumbs on it. Rotate the pans halfway through baking. If you are making muffins, bake them for 15 to 20 minutes, rotating halfway through.
Remove from oven and let the doughnuts cool for 4 to 5 minutes in the pan, then remove them from the pans, and place them on a rack to cool completely before dipping into the maple glaze.
While the doughnuts are cooling, whisk together 2 tablespoons milk, maple syrup, vanilla, and powdered sugar; whisk until completely smooth. Add more milk if consistency is too thick. If you want a thicker consistency, add in a bit more powdered sugar.
Next, set a cooling rack over a baking sheet lined with parchment paper.
Dip the top of the baked donuts into the maple glaze, turning to coat well.
Let the donuts rest on the rack set over the parchment paper until glaze sets.