Preheat the oven to 350°F (175°C). Generously grease two 12-cavity donut pans with butter or baking spray. Set aside.

In a medium saucepan over medium heat, combine the apple cider, 1 teaspoon of cinnamon, and 1 teaspoon of apple pie spice. Bring to a gentle boil, then reduce heat to low and simmer until the cider has reduced to 1 ½ cups (approximately 20-30 minutes). Remove from heat and let cool completely.

In a medium bowl, whisk together the flour, baking powder, baking soda, salt, remaining cinnamon (2 teaspoons), and remaining apple pie spice (2 teaspoons). Set aside.

In a large bowl, whisk together the melted butter, brown sugar, granulated sugar, eggs, milk, and vanilla extract until well combined.

Gradually add the dry ingredients to the wet ingredients, whisking gently until just combined and smooth. Stir in the cooled, concentrated apple cider.

Transfer the batter to a piping bag or zip-top bag. Pipe the batter into each donut cavity, filling them about ¾ full.

Bake for 12-15 minutes, or until the donuts are lightly golden and spring back when touched. Let cool in the pans for a few minutes before transferring them to a wire rack to cool completely.

Inject caramel into each donut using a piping bag fitted with a metal filling tip.

In a shallow bowl, melt the butter. In another bowl, combine granulated sugar, cinnamon, and apple pie spice. Dunk each donut in melted butter, then roll in the cinnamon-sugar mixture to coat.