Place a 1-quart container in the freezer to chill.
Whisk together milk, 1/2 cup cream, sugar, salt, and cinnamon sticks in a medium saucepan. Warm the mixture over medium heat until bubbles begin to form around the edges and steam just begins to rise from the surface. Remove from heat, cover, and let steep for an hour.
Rewarm the cinnamon-infused mixture. Turn off the heat and remove the cinnamon shards with a slotted spoon. It's okay if smaller pieces get left behind.
Set a mesh strainer over a medium bowl. Set aside.
Whisk egg yolks in a small bowl. In several additions, add a some of the warmed cinnamon-infused mixture to the yolks, whisking constantly. Pour the yolk-milk mixture back into the sauce pan. Cook over medium heat, stirring constantly and scraping the bottom of the pan until the mixture begins to thicken.
Remove from heat and immediately pour the custard through the strainer, discarding solids. Whisk in remaining 1 cup heavy cream and the vanilla extract. Cover and refrigerate until the mixture is cold.