Preheat your oven to 375° and place rack in the middle of the oven. Grease a 12-cup muffin pan and set aside.
Pour the apple cider in a small pot and bring to a boil. Reduce the heat to medium and simmer until the liquid is reduced to 1 cup. Set aside to cool.
Using a stand or handheld mixer, mix the oil with the ½ cup white sugar and ¼ cup brown sugar. One at a time, add the eggs,, mixing well after each. Add the vanilla and mix.
In a medium-size bowl, whisk together the flour, 1 teaspoon cinnamon, baking powder, baking soda, and salt. Slowly add the dry ingredients to the wet while the mixer is on low speed. Next, slowly add the reduced cider and beat to combine. Divide the batter evenly among the prepared muffin cups and transfer to the oven. Bake about 15 minutes. Remove from the oven and cool for about 8-10 minutes.
While muffins cool, mix ½ cup white sugar with 1 teaspoon cinnamon. Once cool, brush the muffin with butter on top and sides. Roll in the sugar mixture and set on a drying rack.
Store in a covered container.