In a saucepan, add the cranberry juice and apple cider.
Add the sugar, cinnamon sticks, allspice and nutmeg.
Heat to boiling and then reduce heat, simmering for 10-15 minutes.
Strain to remove spices.
Place into a flat 8" x 8" glass baking dish, or loaf pan.
Freeze for 45 minutes.
Using a fork, lightly scrape to create the crystals.
Repeat this process until it's completely frozen and scraped into loose crystals. It should appear much lighter in color once scraped.
Spoon into chilled decorative glasses with a sprig of mint.
Serve immediately or freeze until ready to enjoy.