In a small saucepan over medium-high heat, simmer the apple cider until it reduces to 0.5 cup, approximately 15 minutes. Set aside to cool.
Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius) and grease a 12-count donut pan with non-stick spray or butter.
In a large bowl, whisk together the flour, baking powder, baking soda, 1 teaspoon cinnamon, nutmeg, and salt until well combined.
In a medium bowl, whisk together the 2 tablespoons melted butter, egg, brown sugar, 0.5 cup granulated sugar, buttermilk, vanilla extract, and cooled reduced apple cider.
Pour the wet ingredients into the dry ingredients and whisk until just combined, ensuring no large lumps of flour remain while avoiding overmixing.
Transfer the batter to a piping bag or a large zip-top bag with the corner snipped off, and pipe the batter into the donut cavities, filling them about two-thirds full.
Bake for 10 to 12 minutes or until the donuts are springy to the touch and a toothpick inserted into the center comes out clean.
Allow the donuts to cool in the pan for 5 minutes before transferring them to a wire rack to cool slightly.
In a small bowl, combine the remaining 0.5 cup granulated sugar and 1 teaspoon cinnamon. Brush each warm donut lightly with the 3 tablespoons of melted butter and toss in the cinnamon sugar mixture until evenly coated.