In a small pot over medium-high heat, bring apple cider to a simmer. Reduce heat to medium and continue to simmer until reduced to about ½ cup, 15 to 20 minutes. Let cool.

Meanwhile, preheat oven to 400°. Generously grease a 24-cup mini muffin tin with butter.

In a medium bowl, whisk flour, baking powder, baking soda, salt, ginger, nutmeg, and ½ teaspoon cinnamon.

In a large bowl, using a handheld mixer on medium-high speed, beat brown sugar and 4 tablespoons butter until light and fluffy, 2 to 3 minutes. Add egg and oil and beat until pale and creamy, about 1 minute more. Add half of dry ingredients and fold with a rubber spatula until combined. Add reduced apple cider, apple butter, and remaining dry ingredients and fold just until combined.

Scoop a heaping tablespoon batter into each prepared cup (cups will be nearly full). Bake muffins until risen and a tester inserted into the center comes out with a few moist crumbs attached, 8 to 10 minutes. Let cool in tin 2 minutes.

Meanwhile, in a wide shallow heatproof dish, microwave remaining 8 tablespoons butter in 15-second increments until melted. In another wide shallow dish, whisk granulated sugar and remaining 4 teaspoons cinnamon.

Working one at a time, roll muffin in melted butter, then immediately in cinnamon sugar. Transfer to a wire rack and let cool completely.

Make Ahead: Apple cider can be reduced 3 days ahead. Let cool. Transfer to an airtight container and refrigerate.