For the cinnamon ice cream: In a medium saucepan over medium-low heat, combine the half-and-half and sugar. Stir it around to combine and allow to warm. Next, add the vanilla extract and cinnamon sticks. Stir the mixture around and heat it up until it's hot but not boiling.
Add the egg yolks to a medium bowl and whisk them vigorously until slightly lighter in color, about 2 minutes.
Remove the cinnamon sticks from the half-and-half mixture and discard. Grab a ladle of the hot half-and-half mixture and drizzle it very slowly into the yolks, whisking constantly. This will temper the egg yolks so that they don't scramble when they're added to the pan. Add another ladle of the hot liquid, and when it's mixed in, pour the egg mixture slowly into the pan with the half-and-half mixture, stirring with a spoon. Stir and cook it until the mixture is thick and coating the spoon, for 2 minutes or so.
Set a fine-mesh strainer over a clean glass bowl and strain the custard. Don't worry if it looks a little lumpy. Add the heavy cream to the bowl and stir to combine. Finally, whisk in the ground cinnamon, and then cover the bowl and refrigerate until completely cool.
Pour the cooled mixture into an ice cream maker and process according to your machine's instructions. When it's finished churning, it will still be in the soft-serve stage. Transfer the mixture to a freezer container and place in the freezer until it's frozen firm, for several hours.
For the cinnamon crisps: Preheat the oven to 350 degrees F.
Combine the sugar and cinnamon in a small bowl. Place the tortillas on a baking sheet and brush both sides with the butter. Sprinkle both sides of the tortillas generously with the cinnamon-sugar mixture.
Bake until golden brown and crisp, about 15 minutes, and then allow them to cool to room temperature. Break into large pieces and set aside.
Serve the ice cream with the cinnamon crisps.