Preheat the oven to 400 degrees.
In a medium bowl, whisk together the flour, baking powder, salt and cinnamon. Combine the melted butter with the apple cider and sugar, then add to the dry ingredients along with the egg and whisk until just combined. Be careful not to over-mix or it will cause the doughnuts to become tough.
Spray a doughnut pan or muffin tin with cooking spray, then carefully spoon the batter into the pan. Bake for 7-10 minutes, or until a toothpick inserted into the center of the doughnuts comes out clean (this will take longer for muffins). Allow to cool for a minute or two before turning out of the pan.
Pour the remaining melted butter into a shallow bowl, then combine the sugar and cinnamon in another. While the doughnuts are still warm, carefully dip them in the butter on both sides, then roll in the cinnamon sugar until coated. These doughnuts are best served when they're still warm.