Preheat oven to 350°F (175°C) and grease a bundt pan thoroughly, coating all sides of the cake pan.
In a small saucepan, reduce the apple cider over medium heat to about 1/2 cup. Let cool slightly.
In a large mixing bowl, whisk together flour, baking powder, baking soda, cinnamon, apple pie spice, and salt.
In another bowl, whisk eggs, sugars, oil, vanilla, apple sauce, apple cider vinegar, and the reduced apple cider.
Gradually add dry ingredients to the wet mixture, stirring just until combined to avoid overmixing the cake batter.
Pour the batter into the prepared bundt pan and bake for 40-45 minutes. Insert a cake tester in the center of the cake—if it comes out clean, it's done.
Allow the cake to cool in the pan for 15 minutes, then invert onto a wire rack. Brush the warm cake with melted butter, then sprinkle with cinnamon sugar.