Make the apple cider syrup.

In a large saucepan over medium heat, stir together the apple cider, agave syrup, sugar, and the cinnamon sticks.

Cook the mixture, stirring often until it comes to a simmer and reduces to a thick syrup, about 20 minutes.

Remove from the heat and stir in the butter.  Set aside to cool.

Make the donuts.

Preheat the oven to 350-degrees.  Spray a standard donut pan with cooking spray and set aside.

In a large mixing bowl, stir together the eggs, both sugars, vanilla, applesauce, and pumpkin puree until well combined.

In a medium bowl, whisk together the flour, cinnamon, pumpkin pie spice, salt, and baking powder.

Add the dry ingredients slowly to the wet ingredients and fold in just until combined.  Don’t over mix the batter.

Spoon into the prepared donut pan until each cavity is half full.  Use a paper towel to wipe any batter that didn’t make it into a cavity.

Bake for 17 minutes or until firm and golden.

Meanwhile, place the stick of melted butter into a shallow bowl.

Mix the sugar and cinnamon together in another shallow bowl.

When the donuts are done, remove them from the oven and invert the donuts onto a piece of parchment paper.

While still hot, dip each donut quickly into the butter, coating both sides, and then into the cinnamon-sugar mixture.  

Place pack onto the parchment paper and allow to cool completely.  Repeat with the remaining donuts.  

Make the glaze.

In a medium bowl, whisk together the powdered sugar and apple cider syrup until smooth, yet still thick enough to get a thick drizzle.  You may need to add more syrup or powdered sugar depending on the thickness of the glaze. Pour the glaze into a spouted measuring cup or into a ziplock bag.  If using the bag, close tightly and snip off the corner with scissors.

Glaze the donuts in a back and forth fashion.  Let set before serving.