Pour apple cider into a saucepan, and bring to a simmer. Simmer until cider is reduced to ⅔ cup. Then remove from what and let cider cool.
Preheat oven to 350°F. Line a 12-cup standard muffin tin with liners.
In a larger mixing bowl, combine flour, sugar, baking powder, and salt.
Melt butter just until melted - but not hot (if it does get hot, let it cool).
Make a well in the center of flour mixture and add egg. Whisk the egg to break it up, but don’t mix it into the batter yet.
Add in cider, milk, vanilla, and butter. Then stir all ingredients together just until combined. Don't overmix.
Spoon batter into 12 paper-lined (or greased) muffin cups.
Bake for 16 to 18 minutes, until a toothpick inserted into the center comes out clean.
Remove from oven and cool on a wire rack.
In shallow dish mix cinnamon and coarse sugar together. Then melt the butter.
Dip or brush muffins with melted butter, and then cinnamon sugar.
Serve and enjoy!