In a medium saucepan placed over medium heat, add the half-and-half and 1 1/2 cups of the heavy cream. Add sugar and whisk until dissolved. Add cinnamon sticks to steep.
While the cream mixture heats, add 2 egg yolks to the remaining 1/2 cup of heavy cream, whisking to combine. When the cream is hot, slowly ladle some of the hot cream into the cream and egg yolks, whisking continuously. This tempers the eggs.
Then slowly pour the egg yolk mixture into the hot cream mixture, while whisking. Cook over medium-low heat until slightly thickened, being careful not to boil the cream mixture. Stir in ground cinnamon and vanilla extract.
Strain the mixture into a heat-proof bowl. Add the cinnamon sticks back into the cream to continue flavoring the cream while it chills. Cover with plastic wrap, pressing the plastic wrap directly on top of the cream mixture. Refrigerate until completely chilled.
Remove the cinnamon sticks and transfer the cream mixture to an ice cream machine and churn until the mixture is the consistency of soft serve. Transfer to a freezer-safe container and freeze until firm, approximately 3-4 hours.
The prep time and cook time does not include the chill or freeze time.