Pour the apple cider into a small saucepan and bring to a boil over medium-high heat, then reduce to a simmer. Cook for 15–20 minutes, stirring occasionally, until reduced to ¼ cup (60ml) — it should be thick, syrupy, and deep amber. Remove from heat and cool to room temperature.
Preheat oven to 350°F (175°C). Generously spray the donut pan with non-stick cooking spray, making sure to coat all cavities and the center posts thoroughly.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, and allspice for at least 30 seconds until the spices are fully incorporated. Set aside.
In a large bowl, whisk together the melted butter, granulated sugar, and brown sugar. Add the egg and whisk for 1 minute until smooth, then stir in the cooled reduced cider, buttermilk, and vanilla until fully combined.
Add the dry ingredients to the wet ingredients and fold gently with a spatula just until no flour streaks remain — do not overmix. The batter should be thick but pourable, similar to muffin batter.
Transfer the batter to a piping bag or zip-top bag with a corner snipped off, and pipe into each donut cavity about ⅔ full. Avoid overfilling as the donuts will rise during baking.
Bake on the center rack for 10–12 minutes until the donuts are lightly golden and spring back when gently touched; a toothpick should come out clean or with just a few moist crumbs. Cool in the pan for 3–4 minutes.
While the donuts cool, stir together the granulated sugar and cinnamon in a shallow bowl until evenly combined. Melt the remaining 3 tablespoons of butter in a small bowl.
Remove the warm donuts from the pan, brush each one generously with melted butter on all sides, then immediately roll in the cinnamon sugar, pressing gently to adhere. Transfer to a wire rack to cool slightly before serving.