In a saucepan, combine the cream and cinnnamon sticks, and bring them to a boil over medium-high heat. Reduce the heat to low, and cook gently for about 10 minutes, until the cream is infused with the cinnamon. Set the pan aside so the cream can cool.
In a bowl, whisk together the eggs, sugar, and vanilla until they are well mixed. Add the cream and cinnamon sticks, and whisk well.
Heat 1 to 2 inches of water in the bottom pan of a double boiler, and transfer custard mixture to the top pan, or set the bowl over a saucepan containing 1 to 2 inches of hot water. Bring the water to a boil, and cook the sauce, stirring constantly, for about 30 minutes or until it thickens, adding more hot water to the bottom pan if necessary. Remove the top pan or the bowl from over the hot water, and let the custard cool.
Strain the cooled custard through a fine-mesh sieve into a glass or ceramic container (do not use plastic), and refrigerate the natillas for at least 4 hours, until it is cool. Stir before serving, adding a little more heavy cream if necessary to smooth the natillas. Divide it among six custard cups or transfer it to a pitcher to use as a sauce. Serve the natillas sprinkled with cinnamon.
This is also delicious served as a sauce for almond cake, and also with fresh raspberries -- !