Preheat oven to 350F. Grease or line a standard 12 muffin tin.
In a small sauce pan, bring the apple and apple cider to a boil. Then reduce heat and simmer for about 10-15 minutes, until the apple is easily smashed with a fork and the cider has reduced. Remove from heat and let it cool for about 5 minutes. Then pulse with an immersion blender or in a food processor. Measure out 1 cup (you might have a little more than this) and mix it with the milk and apple cider syrup. Set aside.
In a medium bowl whisk together the flour, baking powder, baking soda, salt, nutmeg and mace.
With a mixer, using the paddle attachment, beat the butter and both sugars on medium speed until light and fluffy, about 3 minutes. Add the eggs one at a time, beating well after each addition. Add the oil and beat until incorporated.
With the mixer on low, add the flour mixture in three additions, alternating with the apple/milk mixture. Scrape down the bowl as needed. Increase speed to medium for about 20 seconds. Add vanilla and beat once more for about 30 seconds.
This will be quite a loose batter. Pour the batter into the prepared muffin cups. Bake for 20 - 30 minutes.
While the muffins are baking, combine the ingredients for the cinnamon sugar coating.
While the muffins are still warm, coat them with the cinnamon sugar. If you have used muffins liners, then dip or rub the tops in the cinnamon sugar. If you have un-papered muffins, roll and rub the whole muffin in the cinnamon sugar.