Preheat the oven to 350°F, then boil the cider in a small saucepan over medium-high heat for about 16 to 18 minutes, stopping when ¼ cup/60 milliliters remains and cooling it completely before mixing. The liquid becomes darker and more syrupy as it concentrates, with larger, slower-breaking bubbles near the end.

Whisk the flour, baking powder, salt, baking soda, 1 teaspoon cinnamon, ginger, and nutmeg in a medium bowl; separately, use an electric mixer on medium to cream the softened butter and brown sugar in a large bowl for about 3 minutes, scrape the bowl, beat in the egg, apple butter, and cooled cider, then mix in the dry ingredients just until incorporated. The creamed butter should look fluffy, and any slight separation after the fruit additions should disappear into a thick batter once the flour blends in.

Place 2-tablespoon portions on two parchment-lined rimmed baking sheets with about 1 ½ inches between them, bake for 12 to 14 minutes while rotating the sheets halfway through, move the sheets to racks, combine the remaining ½ teaspoon cinnamon with the granulated sugar, then lightly brush the warm tops with melted butter and gently turn the cakes in the sugar before cooling completely. Pull the cakes when their domes look set and spring back lightly under a fingertip, before the edges become dark or dry.

Use an electric mixer on medium to blend the cream cheese and butter in a large bowl until smooth, incorporate the confectioners’ sugar and vanilla, then spread about 2 tablespoons of filling over the flat bottoms of half the completely cooled cakes and cover with the remaining halves. The filling should look creamy and hold its shape between the cakes without melting at the edges.