Heat oven to 350°F and line a large baking sheet with parchment paper and set aside.
In a small saucepan, bring the apple cider to a boil over medium-high heat. Reduce the cider to about ¼ cup, about 15 to 20 minutes. Allow the reduction to cool in the pot or transfer to a heat-safe mug or bowl and place in the refrigerator to speed up the cooling process.
In a medium bowl, whisk together the flour, baking powder, salt, baking soda, 1 teaspoon cinnamon, ginger, and nutmeg. Set aside.
In a large bowl or stand mixer fitted with a paddle attachment, cream the softened butter and brown sugar until light and fluffy, about 3 minutes, scraping the sides of the bowl as needed.
Add the eggs, apple butter, and cooled apple cider reduction to the bowl and beat until thoroughly combined. Your batter may look like it has separated slightly, that's perfectly normal. Add the dry ingredients and mix just until combined, scraping the bowl as needed.
Using a medium cookie scoop, portion the dough into the prepared baking sheet, about 2 inches apart.
Bake for 10 to 14 minutes, rotating halfway through until the cakes are puffed and set and spring back when touched with a finger. Transfer the baking sheet to a wire rack.
In a small bowl, whisk together the remaining ½ teaspoon of cinnamon and the granulated sugar. Brush the tops of the warm cakes with melted butter and gently toss the tops in the sugar mixture, try not to get the flat parts coated or the filling may have a hard time sticking. Set on a wire rack to cool completely.
In a large bowl or stand mixer fitted with a paddle attachment, cream the butter and brown sugar until light and fluffy. Add the powdered sugar a ½ cup at a time, mixing after each addition just until incorporated. Add the vanilla and heavy cream and beat until light and fluffy. Pipe about 2 tablespoons of filling on the flat side of half of the cakes then top them with the other halves.