Simmer the apple cider in a small nonstick saucepan over medium heat until it reduces to about 1/3 cup and becomes syrupy. Remove from the heat and set aside to cool.

In a large mixing bowl, whisk the egg, granulated sugar, brown sugar, melted butter, milk, and vanilla extract until the mixture is smooth and thoroughly combined.

In a separate bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and cardamom so the leavening and spices are evenly distributed.

Add the dry ingredients to the wet ingredients along with the cooled reduced cider. Fold gently just until combined, being careful not to overmix the batter.

Preheat the oven to 350°F. Lightly grease the wells of a donut pan and fill each well about halfway with batter, using a piping bag or spoon for neat filling.

Bake the donuts for 10 minutes, or until the tops spring back when touched and a toothpick inserted in the center comes out clean.

Let the donuts cool in the pan for about 5 minutes, then transfer them carefully to a wire rack.

In a shallow bowl, stir together the remaining 1 cup granulated sugar, 1 teaspoon cinnamon, 1/2 teaspoon nutmeg, and 1/2 teaspoon cardamom. Brush each warm donut with the melted butter, then roll it generously in the spiced sugar.

Serve the donuts immediately while warm for the best flavor and soft texture.