Preheat the oven to 350°F (175°C). Generously grease and flour a 10- to 12-cup Bundt pan, making sure to coat all the grooves well.
Combine the apple cider and chopped Granny Smith apple in a saucepan over medium heat. Simmer until the apple is tender and the cider has reduced slightly, then remove from the heat and let it cool.
Whisk together the flour, baking powder, baking soda, salt, nutmeg, and mace in a medium bowl.
Beat the softened butter, granulated sugar, and brown sugar until light and fluffy. Add the eggs one at a time, mixing well after each addition, then stir in the vanilla.
Gradually add the dry ingredients, alternating with the milk and cooled apple cider mixture. Mix just until the batter comes together, then gently fold in the cooked apple.
Pour the batter into the prepared Bundt pan and smooth the top. Bake until the cake is golden and a toothpick inserted into the center comes out clean.
Let the cake cool in the pan for about 15 to 20 minutes, then carefully turn it out onto a wire rack and allow it to cool completely.
Mix the granulated sugar and cinnamon in a small bowl. Brush the cooled cake all over with melted butter, then coat generously with the cinnamon sugar, gently pressing it onto the surface.
Slice and serve once the coating has set.