Preheat the oven to 375 degrees Fahrenheit and set a rack to the middle position. Lightly grease the muffin tin. In a saucepan, bring the apple cider to a boil over high heat. Reduce the heat slightly and simmer until the liquid is reduced to 1 cup, 10 to 15 minutes. Set aside to cool.
Using a stand or hand-held mixer, cream the butter with the sugar in a large bowl at medium speed until fluffy, about 4 minutes. Add the eggs, one at a time, blending well after each. Add the vanilla and blend.
In a medium bowl, whisk together the flour, nutmeg, salt, baking powder and baking soda. Add a third of this mixture to the butter mixture and beat just to combine. Add half of the reduced cider and beat to combine. Repeat with another third of the flour mixture, then the rest of the cider, then the remaining flour mixture. Divide the batter evenly among the prepared muffin cups and bake until the tops are firm and a tester inserted into the center comes out clean, 15 to 17 minutes. Let cool for 10 minutes.
Prepare the topping: In a medium bowl, whisk together the sugar and cinnamon. As soon as the muffins are cool enough to handle, brush their tops and sides with the melted butter, then roll them in the cinnamon-sugar to coat. Serve warm or at room temperature.

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