Preheat the oven to 350 degrees and lightly spray two (six donut) donut pans, or muffin tins with cooking spray. In a large measuring cup or batter bowl, whisk together flour, baking powder, sea salt, nutmeg, and 1 teaspoon cinnamon.
In the bowl of a stand mixer with a paddle attachment, cream together the brown sugar, 10 tablespoons of room temperature butter, and 1/4 cup white sugar on medium speed, until light and fluffy.
To the creamed butter mixture, add one egg at a time, mixing between each egg. Scrape down the sides of the bowl as needed to fully incorporate the eggs. Then beat in the vanilla extract.
Add the dry ingredients on low speed and mix until the dry mixture is fully incorporated. With the mixer running on low, pour in the apple cider in a slow and steady stream. Continue mixing until the apple cider is fully incorporated. Scrape down the sides and give it a few mixes by hand to ensure you have a smooth, well-mixed batter.
You can spoon the batter into the tins, or you can use a gallon freezer bag, or a pastry bag to pipe the batter into the tins. Spoon the batter into a pastry bag set in a tall cup and cut a 1/2-inch opening in the bottom corner of the bag. Fill the tins 2/3 of the way up. Place the two tins on a single sheet tray and pop the tray in the oven. Bake for 12-15 minutes, until the tops are puffed and golden, and a toothpick comes out clean.
While the donuts are baking, whisk together the remaining 1/2 cup of white sugar and 1 teaspoon of cinnamon. Melt the remaining 6 tablespoons of butter in the microwave.
Get a sheet pan to turn the buttered donuts out and set that nearby. Then, while the donuts are still in the pan, brush the tops with butter (you can do one tin of donuts at a time), then flip the pan over onto that other sheet tray and brush the bottoms with the melted butter.
Dredge the buttered donuts in the cinnamon sugar mixture until well-coated. Set on a tray and repeat the buttering and dredging process with the remaining donuts.