Crush your first set of candy canes by adding the pieces to a food processor or blender and pulsing until you have a nice fine crumb.
Add those to a large bowl. Then add in the heavy cream, milk, sugar, vanilla, and peppermint extracts (Add the food coloring if you want your ice cream more of a pink color) and stir them all together for a couple minutes until the sugar starts to dissolve.
Pour the mixture into a prepared ice cream machine, and let it churn to run for about 25-30 minutes, according to manufactures instructions, until it is a soft serve consistency.
Add the remaining candy canes to a ziptop bag or the food processor, and pulse to get chunky pieces.
Add the candy cane pieces to the ice cream and let it continue churning for a couple more minutes until they are mixed throughout.
Scoop the ice cream into a 2qt container (a 9x5 bread pan works well) and cover it.
Place the ice cream container in the freezer to chill for at least 6 hours, overnight is even better.
Scoop ice cream and enjoy topped with additional candy cane pieces, and hot fudge, if desired.