Place the bowl of your ice cream maker in the freezer the night before.
To make the ice cream base, whisk together ½ cup of coconut milk and cornstarch. Set aside. Place a saucepan over medium heat and add remaining coconut milk, sugars, and salt, stirring until the sugar has completely dissolved. Add milk/cornstarch mixture and bring to a boil, stirring frequently. Once the coconut milk is boiling, stir continuously for 2 minutes, until the ice cream base is thick and cornstarch is completely cooked. Remove from heat and add the vanilla and cinnamon.
Transfer ice cream base to a shallow container and chill for at least 2 hours. For the best results, this should be made the night before.
Churn the ice cream according to the manufacturer's instructions. Once it is thick and creamy and resembles soft serve, add 1/2 the chopped Snickerdoodles. Churn for 30 seconds more.
In a small loaf pan or similarly sized container place about 1/2 of the ice cream and smooth down. Layer with remaining snickerdoodles and top with remaining ice cream. Return to the freezer for another 2 hours.
Thaw 10 minutes before serving, if necessary.