In a loaf pan, mix together the apple cider, lemon juice, and cinnamon. Whisk until well combined. Freeze the mixture for 1 hour.
Take the cider mixture out of the freezer and use a fork to scrape and stir it, preventing it from freezing into a solid block of ice. Once scraped and stirred, return it to the freezer for another hour.
Remove the cider mixture from the freezer again and use a fork to scrape and stir it until it reaches a shaved ice consistency. If it's not frozen enough to your liking, put it back in the freezer for another 45 minutes. Otherwise, spoon it into 2 glasses and garnish each rim with a mini doughnut. Enjoy your frosty treat!