Pour the apple cider into a small saucepan. Bring it to a gentle simmer over medium heat, then reduce the heat to medium-low. Simmer uncovered for 15 to 20 minutes, stirring occasionally, until the cider reduces to ¾ cup. Remove from the heat and let cool to room temperature.
Preheat the oven to 350°F. Generously grease a 10- to 12-cup Bundt pan with baking spray containing flour, or grease with butter and dust with flour. Make sure every ridge and the center tube are coated.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, allspice, and salt.
In a large bowl, whisk together the brown sugar, granulated sugar, eggs, oil, applesauce, vanilla, and ¾ cup cooled reduced apple cider until smooth.
Add the dry ingredients to the wet ingredients. Stir gently with a rubber spatula just until no dry flour remains. Do not overmix.
Pour the batter into the prepared Bundt pan and smooth the top.
Bake for 45 to 55 minutes, until a toothpick inserted into the center comes out clean or with a few moist crumbs.
Let the cake cool in the pan for 10 to 15 minutes. Carefully invert it onto a wire rack placed over a baking sheet or parchment paper.
In a small bowl, combine the granulated sugar, cinnamon, and salt for the topping.
While the cake is still warm, brush the top, sides, and inner ring with melted butter.
Sprinkle and gently press the cinnamon sugar mixture over the buttered cake until evenly coated.
Let the cake cool for at least 30 minutes before slicing and serving.