Heat oven to 350 F. Lightly grease 2 (6 ct) doughnut pans. In a medium bowl, add flour, baking powder, salt, ½ teaspoon of the ground cinnamon and nutmeg and whisk to combine. Set this aside.
In the bowl of a stand mixer using the paddle attachment, beat 5 tablespoons butter, brown sugar, 1 tablespoon maple syrup and 3 tablespoons of granulated sugar on medium speed until light and fluffy. Add the eggs one at a time and mix until well incorporated after each addition, scraping the bowl as necessary. Add in the vanilla extract.
Add the flour mixture and mix on low speed until incorporated. With the mixer on low, add the apple cider in a steady stream. Once the cider is in, mix on medium-low speed for 30 seconds. Be sure to scrape the sides of the bowl.
The easiest, mess-free way to fill the donut pans would be to: fill them about 2/3 of the way using a disposable piping bag or a ziploc bag with a 1/2-inch cut in the corner. Bake for 15 minutes or until golden brown and a toothpick inserted into the center of the thickest part comes out clean. Rotate the pans halfway through baking so that they cook evenly.
While the donuts bake, stir the remaining ⅓ cup granulated sugar and 1 teaspoon cinnamon together in a small bowl. In a separate small bowl, melt the remaining 3 tablespoons butter in the microwave and stir in 1 tablespoon maple syrup. Let the doughnuts cool for 5 minutes after baking, remove them from the pans, and brush with the melted butter/maple syrup mixture and dredge them in the cinnamon sugar while they are still warm. Set aside to let rest for up to 5 minutes and enjoy!