Heat oven to 350 degrees. Lightly grease 2 (6-cavity) doughnut pans (or a 12-cup muffin tin) with nonstick spray. In a medium bowl, add flour, baking powder, salt, 1 teaspoon cinnamon, and nutmeg and whisk to combine.

In the bowl of a stand mixer fitted with the paddle attachment, cream 10 tablespoons butter, brown sugar, and ¼ cup granulated sugar on medium speed until light and fluffy, about 3 to 4 minutes.

Add the eggs one at a time, mixing until well incorporated after each addition. Beat in the vanilla extract.

Add the flour mixture and mix on low speed until incorporated. With the mixer running, add the apple cider in a slow, steady stream and mix until combined. Scrape the bowl well.

Spoon the batter into prepared doughnut pans, filling them about ⅔ of the way.

Bake until evenly golden brown and a toothpick inserted into the center comes out clean, about 12 to 15 minutes, rotating the pans halfway through.

While the doughnuts bake, whisk the remaining ½ cup granulated sugar and 1 teaspoon cinnamon together in a small bowl.

In another small bowl, melt the remaining 6 tablespoons butter in the microwave.

Let the doughnuts cool for 5 minutes, then unmold, brush with melted butter, and dredge in the cinnamon sugar while still warm.