Mix together 2 cups of the milk, the rice and the cinnamon stick in a medium
saucepan. Bring to a boil and simmer over medium-low heat until the rice is tender,
40 to 45 minutes.
Discard the cinnamon stick and set the pan aside.
Whisk the remaining 1/2 cup milk with the egg yolks in a large bowl.
Add a little of the rice mixture at a time, whisking constantly. When all the rice has been added,
return the pudding to the pan and cook over low heat, stirring, until the temperature
of the pudding reaches 160 degrees F, for 3 to 5 minutes.
Remove from the heat and stir
in the dulce de leche and vanilla extract. Stir until incorporated.
Pour the pudding into a bowl, cover and refrigerate until cold, at least 4 hours or overnight.