Preheat the oven to 425°F and lightly spray the mini doughnut pan with non stick cooking spray.
For the glaze: Heat the cider and rum in a small saucepan over medium-high heat. Simmer for about 15-20 minutes or until the liquid is reduced to approximately 1/4 cup. Whisk in the confectioners’ sugar until no lumps remain. Whisk in butter and set aside.
For the doughnuts: Whisk together the flour, baking powder, salt, pumpkin pie spice in a medium bowl. In a separate large bowl, whisk the egg with the pumpkin purée, brown sugar, buttermilk butter and vanilla. Add the dry ingredients and mix gently until just combined. Transfer to a resealable plastic bag or a pastry bag. Cut off the corner of the bag and pipe the batter into the doughnut mold.
Bake for 6 to 7 minutes or until springy to the touch. Let cool slightly (1-3 minutes) before transferring to a wire rack with parchment paper or baking sheet underneath. Spoon the glaze over the top of the doughnuts. (Or dip them for even better coverage!)
If you don’t have buttermilk, make a substitute: Put 1 teaspoon white vinegar in a measuring cup. Add milk to make 1/3 cup. Stir and let sit for 10 minutes. Then it is ready to use!