Pour the apple cider into a small saucepan and bring to a gentle boil over medium heat. Reduce to a simmer and cook until the cider has reduced to about 1/2 cup, approximately 20-25 minutes. Allow to cool completely before using.
In a large bowl, whisk together the flour, brown sugar, baking powder, baking soda, cinnamon, nutmeg, salt, allspice, and cardamom (if using).
In a separate medium bowl, whisk the eggs, melted butter, buttermilk, cooled reduced apple cider, and vanilla extract until well combined.
Pour the wet ingredients into the dry ingredients and mix just until combined. Be careful not to overmix - stop when the flour is just incorporated.
Cover the dough with plastic wrap, pressing it directly onto the surface. Refrigerate for at least 1 hour or up to overnight.
Preheat your oven to 350°F (175°C). Grease two doughnut pans with non-stick cooking spray or butter.
Transfer the batter to a large piping bag or zip-top bag with a corner snipped off. Pipe the batter into the prepared doughnut pans, filling each cavity about 3/4 full. Tap the pans gently on the counter to remove any air bubbles.
Bake for 15-18 minutes, or until the tops spring back when lightly touched and a toothpick inserted comes out clean.
While the doughnuts are baking, mix the granulated sugar and cinnamon in a shallow bowl. Melt the butter in another bowl.
Allow the doughnuts to cool in the pan for 5 minutes before carefully removing. While still warm, brush each doughnut with melted butter and then roll in the cinnamon-sugar mixture to coat completely.