Heat oven to 350 degrees. Lightly grease 2 6-cavity doughnut pans (or a 12-cup muffin pan) with nonstick spray.
In a medium bowl, add gf flour, baking powder, salt, 1 teaspoon cinnamon and nutmeg and whisk to combine. Set aside.
In bowl of a stand mixer fitted with the paddle attachment, cream 10 tablespoons butter, brown sugar and 1/4 cup/50 grams granulated sugar on medium speed until light and fluffy, 3 to 4 minutes. Add the eggs one at a time and mix until well incorporated after each addition, scraping bowl as necessary. Beat in vanilla extract.
Slowly add flour mixture to the batter and mix on low speed until incorporated.
With the mixer running, carefully add apple cider in a slow, steady stream. Scrape the bowl well to make sure the batter is thoroughly incorporated.
Spoon batter into a plastic baggie (or pastry bag). Cut a 1/2 -inch opening in the bottom corner tip of baggie and squeeze batter into doughnut pans, filling them about 2/3 of the way full. (If you don't own doughnut pans, you can use a muffin pan and either make muffins or add a small, squished, rounded piece of foil in the middle of each cup to simulate a doughnut pan as I did above.)
Bake until evenly golden brown and a toothpick inserted into the center of the thickest portion comes out clean, 12 to 15 minutes for doughnuts or 15-20 minutes for muffins. Rotate pan(s) halfway through baking.
While the doughnuts bake, whisk remaining 1/2 cup/100 grams granulated sugar and 1 teaspoon cinnamon together in a small bowl to combine.
In a separate small bowl, melt remaining 6 tablespoons of butter on the stove or in the microwave.
Let doughnuts cool for 5 minutes after baking, then unmold them from the pans, brush all sides with the melted butter and dredge them in cinnamon sugar mixture while they are still warm.
Serve immediately or let cool to room temperature.