In a saucepan placed over medium heat, add the half-and-half and 1 ⅓ cups of the heavy cream. Add sugar and whisk until dissolved. Add cinnamon sticks to steep. Heat the mixture for 4-5 minutes.
While the cream mixture heats, add the egg yolks to the remaining ½ cup of heavy cream, whisking to combine. When the cream is hot, slowly ladle some of the hot cream into the egg yolk mixture, whisking continuously. This tempers the eggs. (Note 1)
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.
Remove from the heat. Stir in ground cinnamon and vanilla extract. Let cool for 10-15 minutes. The cinnamon sticks will continue to infuse the cream.
Strain the mixture into a heat-proof bowl. Cover with plastic wrap, pressing the plastic wrap directly on top of the cream mixture. (Note 2) Refrigerate until completely chilled, 1-2 hours.
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.
Tempering is the process of slowly bringing the eggs up to the temperature of the warm cream. This helps to prevent the eggs from scrambling when they are added to the hot cream.
When chilling a custard (a mixture of cream and eggs), press the plastic wrap down directly on top of the mixture. This prevents a skin from forming on top.