In a saucepan, combine white rice, 2 1/2 cups water, 1 1/2 cups milk and cinnamon stick; bring to boil. Cook for 20 minutes, stirring occasionally, or until rice is tender and most of the liquid is absorbed. Remove cinnamon stick.
Stir in evaporated milk, remaining milk, sugar and vanilla. Return to boil; cook, stirring often, for 5 minutes.
Transfer mixture to a blender, in batches if needed; purée until smooth.
Divide among 8 mugs. Garnish each serving with a pinch of cocoa and cinnamon.
If desired, drizzle with dulce de leche or chocolate syrup before serving.